diff --git a/changelog b/changelog index 02caa58..352a0cf 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,6 @@ +20081208 tpd src/axiom-website/patches.html 20081208.01.tpd.patch +20081208 tpd src/input/Makefile hyperbolicrules.input added +20081208 tpd src/input/hyperbolicrules.input added 20081206 tpd src/axiom-website/patches.html 20081206.01.tpd.patch 20081206 tpd src/input/Makefile add fixed.regress 20081206 tpd src/input/fixed.input convert to a regression file diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index 2428e91..2e7c231 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -787,6 +787,8 @@ bug 7161: integer.spad remove signature change
schaum17 fix 14.355, 14.356
20081206.01.tpd.patch regression file fixed created
+20081208.01.tpd.patch +CATS hyperbolicrules.input added
\ No newline at end of file diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index 2cf453c..45343ce 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -314,7 +314,8 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress alist.regress \ gbf.regress genups.regress gonshor.regress grpthry.regress \ gstbl.regress heap.regress heat.regress help.regress \ herm.regress heugcd.regress \ - hexadec.regress ico.regress ideal.regress \ + hexadec.regress hyperbolicrules.regress \ + ico.regress ideal.regress \ ifact.regress ifthenelse.regress \ infprod.regress intaf.regress intbypart.regress \ intdeq.regress \ @@ -582,7 +583,8 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input ${OUT}/algfacob.input \ ${OUT}/gstbl.input \ ${OUT}/heap.input ${OUT}/heat.input ${OUT}/helix.input \ ${OUT}/herm.input ${OUT}/heugcd.input \ - ${OUT}/hexadec.input ${OUT}/huang.input \ + ${OUT}/hexadec.input ${OUT}/huang.input \ + ${OUT}/hyperbolicrules.input \ ${OUT}/ico.input ${OUT}/ideal.input ${OUT}/ifact.input \ ${OUT}/ifthenelse.input \ ${OUT}/images1.input ${OUT}/images1a.input ${OUT}/images3a.input \ @@ -875,7 +877,8 @@ DOCFILES= \ ${DOC}/help.input.dvi ${DOC}/herm.input.dvi \ ${DOC}/heugcd.input.dvi \ ${DOC}/hexadec.input.dvi ${DOC}/hilbert.as.dvi \ - ${DOC}/huang.input.dvi ${DOC}/ico.input.dvi \ + ${DOC}/huang.input.dvi ${DOC}/hyperbolicrules.input.dvi \ + ${DOC}/ico.input.dvi \ ${DOC}/ideal.input.dvi ${DOC}/ifact.input.dvi \ ${DOC}/ifthenelse.input.dvi \ ${DOC}/images1a.input.dvi ${DOC}/images1.input.dvi \ diff --git a/src/input/hyperbolicrules.input.pamphlet b/src/input/hyperbolicrules.input.pamphlet new file mode 100644 index 0000000..fd3af28 --- /dev/null +++ b/src/input/hyperbolicrules.input.pamphlet @@ -0,0 +1,3130 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/input hyperbolicrules.input} +\author{Timothy Daly} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\tableofcontents +\eject +\section{Definition of Hyperbolic Functions} +\subsection{8.1 Hyperbolic sine of x} +\[{\rm sinh}(x) == \frac{e^x-e^{-x}}{2}\] +<<*>>= +)spool hyperbolicrules.output +)set message test on +)set message auto off +)clear all + +--S 1 of 298 +sinhdef:=rule(sinh(x) == (e^x-e^(-x))/2) +--R +--R x - x +--R e - e +--R (1) sinh(x) == --------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 1 + +--S 2 of 298 +t1:=sinh(x) - (e^x-e^(-x))/2 +--R +--R x - x +--R - e + e + 2sinh(x) +--R (2) ---------------------- +--R 2 +--R Type: Expression Integer +--E 2 + +--S 3 of 298 +t2:=sinhdef t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 3 + +@ +\subsection{8.2 Hyperbolic cosine of x} +\[{\rm cosh}(x) == \frac{e^x+e^{-x}}{2}\] +<<*>>= +)clear all + +--S 4 of 298 +coshdef:=rule(cosh(x) == (e^x+e^(-x))/2) +--R +--R x - x +--R e + e +--R (1) cosh(x) == --------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 4 + +--S 5 of 298 +t1:=cosh(x) - (e^x+e^(-x))/2 +--R +--R x - x +--R - e - e + 2cosh(x) +--R (2) ---------------------- +--R 2 +--R Type: Expression Integer +--E 5 + +--S 6 of 298 +t2:=coshdef t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 6 + +@ +\subsection{8.3 Hyperbolic tangent of x} +\[{\rm tanh}(x) == \frac{e^x-e^{-x}}{e^x+e^{-x}}\] +<<*>>= +)clear all + +--S 7 of 298 +tanhdef:=rule(tanh(x) == (e^x-e*(-x))/(e^x+e*(-x))) +--R +--R x +--R e + e x +--R (1) tanh(x) == -------- +--R x +--R e - e x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 7 + +--S 8 of 298 +t1:=tanh(x) - (e^x-e*(-x))/(e^x+e*(-x)) +--R +--R x +--R (tanh(x) - 1)e - e x tanh(x) - e x +--R (2) ----------------------------------- +--R x +--R e - e x +--R Type: Expression Integer +--E 8 + +--S 9 of 298 +t2:=tanhdef t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 9 + +@ +\subsection{8.4 Hyperbolic cotangent of x} +\[{\rm coth}(x) == \frac{e^x+e^{-x}}{e^x-e^{-x}}\] +<<*>>= +)clear all + +--S 10 of 298 +cothdef:=rule(coth(x) == (e^x+e*(-x))/(e^x-e*(-x))) +--R +--R x +--R e - e x +--R (1) coth(x) == -------- +--R x +--R e + e x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 10 + +--S 11 of 298 +t1:=coth(x) - (e^x+e*(-x))/(e^x-e*(-x)) +--R +--R x +--R (coth(x) - 1)e + e x coth(x) + e x +--R (2) ----------------------------------- +--R x +--R e + e x +--R Type: Expression Integer +--E 11 + +--S 12 of 298 +t2:=cothdef t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 12 + +@ +\subsection{8.5 Hyperbolic secant of x} +\[{\rm sech}(x) == \frac{2}{e^x+e^{-x}}\] +<<*>>= +)clear all + +--S 13 of 298 +sechdef:=rule(sech(x) == 2/(e^x+e*(-x))) +--R +--R 2 +--R (1) sech(x) == -------- +--R x +--R e - e x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 13 + +--S 14 of 298 +t1:=sech(x) - 2/(e^x+e*(-x)) +--R +--R x +--R sech(x)e - e x sech(x) - 2 +--R (2) --------------------------- +--R x +--R e - e x +--R Type: Expression Integer +--E 14 + +--S 15 of 298 +t2:=sechdef t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 15 + +@ +\subsection{8.6 Hyperbolic cosecant of x} +\[{\rm csch}(x) == \frac{2}{e^x-e^{-x}}\] +<<*>>= +)clear all + +--S 16 of 298 +cschdef:=rule(csch(x) == 2/(e^x-e*(-x))) +--R +--R 2 +--R (1) csch(x) == -------- +--R x +--R e + e x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 16 + +--S 17 of 298 +t1:=csch(x) - 2/(e^x-e*(-x)) +--R +--R x +--R csch(x)e + e x csch(x) - 2 +--R (2) --------------------------- +--R x +--R e + e x +--R Type: Expression Integer +--E 17 + +--S 18 of 298 +t2:=cschdef t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 18 + +@ +\section{Relationship among Hyperbolic Functions} +\subsection{8.7 ${\rm tanh}(x)$} +\[{\rm tanh}(x) == \frac{{\rm sinh}(x)}{{\rm cosh}(x)}\] +<<*>>= +)clear all + +--S 19 of 298 +tanhrule:=rule(tanh(x) == sinh(x)/cosh(x)) +--R +--R sinh(x) +--R (1) tanh(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 19 + +--S 20 of 298 +t1:=tanh(x) - sinh(x)/cosh(x) +--R +--R cosh(x)tanh(x) - sinh(x) +--R (2) ------------------------ +--R cosh(x) +--R Type: Expression Integer +--E 20 + +--S 21 of 298 +t2:=tanhrule t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 21 + +@ +\subsection{8.8 ${\rm coth}(x)$} +\[{\rm coth}(x) == \frac{1}{{\rm tanh}(x)}\] +\[== \frac{{\rm cosh}(x)}{{\rm sinh}(x)}\] +<<*>>= +)clear all + +--S 22 of 298 +cothrule:=rule(coth(x) == 1/tanh(x)) +--R +--R 1 +--R (1) coth(x) == ------- +--R tanh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 22 + +--S 23 of 298 +t1:=coth(x) - 1/tanh(x) +--R +--R coth(x)tanh(x) - 1 +--R (2) ------------------ +--R tanh(x) +--R Type: Expression Integer +--E 23 + +--S 24 of 298 +t2:=cothrule t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 24 + +--S 25 of 298 +cothrule2:=rule(coth(x) == cosh(x)/sinh(x)) +--R +--R cosh(x) +--R (4) coth(x) == ------- +--R sinh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 25 + +--S 26 of 298 +t3:=coth(x) - cosh(x)/sinh(x) +--R +--R coth(x)sinh(x) - cosh(x) +--R (5) ------------------------ +--R sinh(x) +--R Type: Expression Integer +--E 26 + +--S 27 of 298 +t4:=cothrule2 t3 +--R +--R (6) 0 +--R Type: Expression Integer +--E 27 + +@ +\subsection{8.9 ${\rm sech}(x)$} +\[{\rm sech}(x) == \frac{1}{{\rm cosh}(x)}\] +<<*>>= +)clear all + +--S 28 of 298 +sechrule:=rule(sech(x) == 1/cosh(x)) +--R +--R 1 +--R (1) sech(x) == ------- +--R cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 28 + +--S 29 of 298 +t1:=sech(x) - 1/cosh(x) +--R +--R cosh(x)sech(x) - 1 +--R (2) ------------------ +--R cosh(x) +--R Type: Expression Integer +--E 29 + +--S 30 of 298 +t2:=sechrule t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 30 + +@ +\subsection{8.10 ${\rm csch}(x)$} +\[{\rm csch}(x) == \frac{1}{{\rm sinh}(x)}\] +<<*>>= +)clear all + +--S 31 of 298 +cschrule:=rule(csch(x) == 1/sinh(x)) +--R +--R 1 +--R (1) csch(x) == ------- +--R sinh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 31 + +--S 32 of 298 +t1:=csch(x) - 1/sinh(x) +--R +--R csch(x)sinh(x) - 1 +--R (2) ------------------ +--R sinh(x) +--R Type: Expression Integer +--E 32 + +--S 33 of 298 +t2:=cschrule t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 33 + +@ +\subsection{8.11 ${\rm cosh}^2(x)-~{\rm sinh}^2(x)$} +\[{\rm cosh}^2(x)-~{\rm sinh}^2(x) == 1\] +<<*>>= +)clear all + +--S 34 of 298 +coshsinh:=rule(cosh(x)^2-sinh(x)^2 == 1) +--R +--R 2 2 +--I (1) - sinh(x) + cosh(x) + %Y == %Y + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 34 + +--S 35 of 298 +t1:=cosh(x)^2-sinh(x)^2 - 1 +--R +--R 2 2 +--R (2) - sinh(x) + cosh(x) - 1 +--R Type: Expression Integer +--E 35 + +--S 36 of 298 +t2:=coshsinh t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 36 + +@ +\subsection{8.12 ${\rm sech}^2(x)+~{\rm tanh}^2(x)$} +\[{\rm sech}^2(x)+~{\rm tanh}^2(x) == 1\] +<<*>>= +)clear all + +--S 37 of 298 +sechtanh:=rule(sech(x)^2+tanh(x)^2 == 1) +--R +--R 2 2 +--I (1) tanh(x) + sech(x) + %Z == %Z + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 37 + +--S 38 of 298 +t1:=sech(x)^2+tanh(x)^2 - 1 +--R +--R 2 2 +--R (2) tanh(x) + sech(x) - 1 +--R Type: Expression Integer +--E 38 + +--S 39 of 298 +t2:=sechtanh t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 39 + +@ +\subsection{8.13 ${\rm coth}^2(x)-~{\rm csch}^2(x)$} +\[{\rm coth}^2(x)-~{\rm csch}^2(x) == 1\] +<<*>>= +)clear all + +--S 40 of 298 +cothcsch:=rule(coth(x)^2-csch(x)^2 == 1) +--R +--R 2 2 +--I (1) - csch(x) + coth(x) + %BA == %BA + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 40 + +--S 41 of 298 +t1:=coth(x)^2-csch(x)^2 - 1 +--R +--R 2 2 +--R (2) - csch(x) + coth(x) - 1 +--R Type: Expression Integer +--E 41 + +--S 42 of 298 +t2:=cothcsch t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 42 + +@ +\section{Functions of Negative Arguments} +Axiom already knows these simplifications +\subsection{8.14 ${\rm sinh}(-x)$} +\[{\rm sinh}(-x) == -~{\rm sinh}(x)\] +<<*>>= +)clear all + +--S 43 of 298 +sinh(-x) +--R +--R (1) - sinh(x) +--R Type: Expression Integer +--E 43 + + +@ +\subsection{8.15 ${\rm cosh}(-x)$} +\[{\rm cosh}(-x) == {\rm cosh}(x)\] +<<*>>= +)clear all + +--S 44 of 298 +cosh(-x) +--R +--R (1) cosh(x) +--R Type: Expression Integer +--E 44 + +@ +\subsection{8.16 ${\rm tanh}(-x)$} +\[{\rm tanh}(-x) == -~{\rm tanh}(x)\] +<<*>>= +)clear all + +--S 45 of 298 +tanh(-x) +--R +--R (1) - tanh(x) +--R Type: Expression Integer +--E 45 + +@ +\subsection{8.17 ${\rm csch}(-x)$} +\[{\rm csch}(-x) == -~{\rm csch(x)}\] +<<*>>= +)clear all + +--S 46 of 298 +csch(-x) +--R +--R (1) - csch(x) +--R Type: Expression Integer +--E 46 + +@ +\subsection{8.18 ${\rm sech}(-x)$} +\[{\rm sech}(-x) == {\rm sech}(x)\] +<<*>>= +)clear all + +--S 47 of 298 +sech(-x) +--R +--R (1) sech(x) +--R Type: Expression Integer +--E 47 + +@ +\subsection{8.19 ${\rm coth}(-x)$} +\[{\rm coth}(-x) == -~{\rm coth}(x)\] +<<*>>= +)clear all + +--S 48 of 298 +coth(-x) +--R +--R (1) - coth(x) +--R Type: Expression Integer +--E 48 + +@ +\section{Addition Formulas} +\subsection{8.20 ${\rm sinh}(x \pm y)$} +\[{\rm sinh}(x \pm y) == {\rm sinh}(x)~{\rm cosh}(y)\pm~{\rm cosh}(x)~{\rm sinh}(y)\] +<<*>>= +)clear all + +--S 49 of 298 +sinhadd:=rule(sinh(x+y) == sinh(x)*cosh(y)+cosh(x)*sinh(y)) +--R +--R (1) sinh(y + x) == cosh(x)sinh(y) + cosh(y)sinh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 49 + +--S 50 of 298 +t1:=sinh(x+y) - (sinh(x)*cosh(y)+cosh(x)*sinh(y)) +--R +--R (2) sinh(y + x) - cosh(x)sinh(y) - cosh(y)sinh(x) +--R Type: Expression Integer +--E 50 + +--S 51 of 298 +t2:=sinhadd t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 51 + +--S 52 of 298 +sinhsub:=rule(sinh(x-y) == sinh(x)*cosh(y)-cosh(x)*sinh(y)) +--R +--I (4) - %T sinh(y - x) == - %T cosh(x)sinh(y) + %T cosh(y)sinh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 52 + +--S 53 of 298 +t3:=sinh(x-y) - (sinh(x)*cosh(y)-cosh(x)*sinh(y)) +--R +--R (5) cosh(x)sinh(y) - sinh(y - x) - cosh(y)sinh(x) +--R Type: Expression Integer +--E 53 + +--S 54 of 298 +t4:=sinhsub t3 +--R +--R (6) 0 +--R Type: Expression Integer +--E 54 + +@ +\subsection{8.21 ${\rm cosh}(x \pm y)$} +\[{\rm cosh}(x \pm y) == {\rm cosh}(x)~{\rm cosh}(y)\pm~{\rm sinh}(x)~{\rm sinh}(y)\] +<<*>>= +)clear all + +--S 55 of 298 +coshadd:=rule(cosh(x+y) == cosh(x)*cosh(y)+sinh(x)*sinh(y)) +--R +--R (1) cosh(y + x) == sinh(x)sinh(y) + cosh(x)cosh(y) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 55 + +--S 56 of 298 +t1:=cosh(x+y) - (cosh(x)*cosh(y)+sinh(x)*sinh(y)) +--R +--R (2) - sinh(x)sinh(y) + cosh(y + x) - cosh(x)cosh(y) +--R Type: Expression Integer +--E 56 + +--S 57 of 298 +t2:=coshadd t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 57 + +--S 58 of 298 +coshsub:=rule(cosh(x-y) == cosh(x)*cosh(y)-sinh(x)*sinh(y)) +--R +--R (4) cosh(y - x) == - sinh(x)sinh(y) + cosh(x)cosh(y) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 58 + +--S 59 of 298 +t3:=cosh(x-y) - (cosh(x)*cosh(y)-sinh(x)*sinh(y)) +--R +--R (5) sinh(x)sinh(y) - cosh(x)cosh(y) + cosh(y - x) +--R Type: Expression Integer +--E 59 + +--S 60 of 298 +t4:=coshsub t3 +--R +--R (6) 0 +--R Type: Expression Integer +--E 60 + +@ +\subsection{8.22 ${\rm tanh}(x \pm y)$} +\[{\rm tanh}(x \pm y) == \frac{{\rm tanh}(x)\pm~{\rm tanh}(y)}{1\pm~{\rm tanh}(x)~{\rm tanh}(y)}\] +<<*>>= +)clear all + +--S 61 of 298 +tanhadd:=rule(tanh(x+y) == (tanh(x)+tanh(y))/(1+tanh(x)*tanh(y))) +--R +--R tanh(y) + tanh(x) +--R (1) tanh(y + x) == ------------------ +--R tanh(x)tanh(y) + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 61 + +--S 62 of 298 +t1:=tanh(x+y) - (tanh(x)+tanh(y))/(1+tanh(x)*tanh(y)) +--R +--R (tanh(x)tanh(y) + 1)tanh(y + x) - tanh(y) - tanh(x) +--R (2) --------------------------------------------------- +--R tanh(x)tanh(y) + 1 +--R Type: Expression Integer +--E 62 + +--S 63 of 298 +t2:=tanhadd t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 63 + +--S 64 of 298 +tanhneg:=rule(tanh(x-y) == (tanh(x)-tanh(y))/(1-tanh(x)*tanh(y))) +--R +--I %V tanh(y) - %V tanh(x) +--I (4) - %V tanh(y - x) == ----------------------- +--R tanh(x)tanh(y) - 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 64 + +--S 65 of 298 +t3:=tanh(x-y) - (tanh(x)-tanh(y))/(1-tanh(x)*tanh(y)) +--R +--R (- tanh(x)tanh(y - x) - 1)tanh(y) + tanh(y - x) + tanh(x) +--R (5) --------------------------------------------------------- +--R tanh(x)tanh(y) - 1 +--R Type: Expression Integer +--E 65 + +@ +This loops forever. +<<*>>= +--S 66 of 298 +-- t4:=tanhneg t3 +--E 66 + +@ +\subsection{8.23 ${\rm coth}(x \pm y)$} +\[{\rm coth}(x \pm y) == \frac{{\rm coth}(x)~{\rm coth}(y)\pm 1}{{\rm coth}(y)\pm~{\rm coth}(x)}\] +<<*>>= +)clear all + +--S 67 of 298 +cothadd:=rule(coth(x+y) == (coth(x)*coth(y)+1)/(coth(y)+coth(x))) +--R +--R coth(x)coth(y) + 1 +--R (1) coth(y + x) == ------------------ +--R coth(y) + coth(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 67 + +--S 68 of 298 +t1:=coth(x+y) - (coth(x)*coth(y)+1)/(coth(y)+coth(x)) +--R +--R (coth(y) + coth(x))coth(y + x) - coth(x)coth(y) - 1 +--R (2) --------------------------------------------------- +--R coth(y) + coth(x) +--R Type: Expression Integer +--E 68 + +--S 69 of 298 +t2:=cothadd t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 69 + +--S 70 of 298 +cothneg:=rule(coth(x-y) == (coth(x)*coth(y)-1)/(coth(y)-coth(x))) +--R +--I %W coth(x)coth(y) - %W +--I (4) - %W coth(y - x) == ---------------------- +--R coth(y) - coth(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 70 + +--S 71 of 298 +t3:=coth(x-y) - (coth(x)*coth(y)-1)/(coth(y)-coth(x)) +--R +--R (- coth(y - x) - coth(x))coth(y) + coth(x)coth(y - x) + 1 +--R (5) --------------------------------------------------------- +--R coth(y) - coth(x) +--R Type: Expression Integer +--E 71 + +@ +This loops forever +<<*>>= +--S 72 of 298 +--t4:=cothneg t3 +--E 72 + +@ +\section{Double Angle Formulas} +\subsection{8.24 ${\rm sinh}(2x)$} +\[{\rm sinh}(2x) == 2~{\rm sinh}(x)~{\rm cosh}(x)\] +<<*>>= +)clear all + +--S 73 of 298 +sinh2x:=rule(sinh(2*x) == 2*sinh(x)*cosh(x)) +--R +--R (1) sinh(2x) == 2cosh(x)sinh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 73 + +--S 74 of 298 +t1:=sinh(2*x) - 2*sinh(x)*cosh(x) +--R +--R (2) sinh(2x) - 2cosh(x)sinh(x) +--R Type: Expression Integer +--E 74 + +--S 75 of 298 +t2:=sinh2x t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 75 + +@ +\subsection{8.25 ${\rm cosh}(2x)$} +\[{\rm cosh}(2x) == {\rm cosh}^2(x)+~{\rm sinh}^2(x)\] +\[==2~{\rm cosh}^2(x)-1\] +\[==1+2~{\rm sinh}^2(x)\] +<<*>>= +)clear all + +--S 76 of 298 +cosh2x:=rule(cosh(2*x) == cosh(x)^2+sinh(x)^2) +--R +--R 2 2 +--R (1) cosh(2x) == sinh(x) + cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 76 + +--S 77 of 298 +t1:=cosh(2*x) - (cosh(x)^2+sinh(x)^2) +--R +--R 2 2 +--R (2) - sinh(x) + cosh(2x) - cosh(x) +--R Type: Expression Integer +--E 77 + +--S 78 of 298 +t2:=cosh2x t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 78 + +--S 79 of 298 +cosh2x2:=rule(cosh(2*x) == 2*cosh(x)^2-1) +--R +--R 2 +--R (4) cosh(2x) == 2cosh(x) - 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 79 + +--S 80 of 298 +t3:=cosh(2*x) - (2*cosh(x)^2-1) +--R +--R 2 +--R (5) cosh(2x) - 2cosh(x) + 1 +--R Type: Expression Integer +--E 80 + +--S 81 of 298 +t4:=cosh2x2 t3 +--R +--R (6) 0 +--R Type: Expression Integer +--E 81 + +--S 82 of 298 +cosh2x3:=rule(cosh(2*x) == 1+2*sinh(x)^2) +--R +--R 2 +--R (7) cosh(2x) == 2sinh(x) + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 82 + +--S 83 of 298 +t5:=cosh(2*x) - (1+2*sinh(x)^2) +--R +--R 2 +--R (8) - 2sinh(x) + cosh(2x) - 1 +--R Type: Expression Integer +--E 83 + +--S 84 of 298 +t6:=cosh2x3 t5 +--R +--R (9) 0 +--R Type: Expression Integer +--E 84 + +@ +\subsection{8.26 ${\rm tanh}(2x)$} +\[{\rm tanh}(2x) == \frac{2~{\rm tanh}(x)}{1+~{\rm tanh}^2(x)}\] +<<*>>= +)clear all + +--S 85 of 298 +tanh2x:=rule(tanh(2*x) == (2*tanh(x))/(1+tanh(x)^2)) +--R +--R 2tanh(x) +--R (1) tanh(2x) == ------------ +--R 2 +--R tanh(x) + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 85 + +--S 86 of 298 +t1:=tanh(2*x) - (2*tanh(x))/(1+tanh(x)^2) +--R +--R 2 +--R (tanh(x) + 1)tanh(2x) - 2tanh(x) +--R (2) --------------------------------- +--R 2 +--R tanh(x) + 1 +--R Type: Expression Integer +--E 86 + +--S 87 of 298 +t2:=tanh2x t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 87 + +@ +\section{Half Angle Formulas} +\subsection{8.27 ${\rm sinh}\left(\frac{x}{2}\right)$} +\[{\rm sinh}\left(\frac{x}{2}\right) == \pm\sqrt{\frac{{\rm cosh}(x) - 1}{2}}\quad [+~{\rm if\ \ }x > 0, -{\rm if\ \ }x < 0]\] + +if $x > 0$ then +<<*>>= +)clear all + +--S 88 of 298 +sinhhalf:=rule(sinh(x/2) == sqrt((cosh(x)-1)/2)) +--R +--R +-----------+ +--R x \|cosh(x) - 1 +--R (1) sinh(-) == -------------- +--R 2 +-+ +--R \|2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 88 + +--S 89 of 298 +t1:=sinh(x/2) - sqrt((cosh(x)-1)/2) +--R +--R +-----------+ +-+ x +--R - \|cosh(x) - 1 + \|2 sinh(-) +--R 2 +--R (2) ------------------------------ +--R +-+ +--R \|2 +--R Type: Expression Integer +--E 89 + +--S 90 of 298 +t2:=sinhhalf t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 90 + +@ +if $x < 0$ then +<<*>>= +)clear all + +--S 91 of 298 +sinhhalfneg:=rule(sinh(x/2) == -sqrt((cosh(x)-1)/2)) +--R +--R +-----------+ +--R x \|cosh(x) - 1 +--R (1) sinh(-) == - -------------- +--R 2 +-+ +--R \|2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 91 + +--S 92 of 298 +t1:=sinh(x/2) - -sqrt((cosh(x)-1)/2) +--R +--R +-----------+ +-+ x +--R \|cosh(x) - 1 + \|2 sinh(-) +--R 2 +--R (2) ---------------------------- +--R +-+ +--R \|2 +--R Type: Expression Integer +--E 92 + +--S 93 of 298 +t2:=sinhhalfneg t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 93 + +@ +\subsection{8.28 ${\rm cosh}\left(\frac{x}{2}\right)$} +\[{\rm cosh}\left(\frac{x}{2}\right) == \sqrt{\frac{{\rm cosh}(x) + 1}{2}}\] +<<*>>= +)clear all + +--S 94 of 298 +coshhalf:=rule(cosh(x/2) == sqrt((cosh(x)+1)/2)) +--R +--R +-----------+ +--R x \|cosh(x) + 1 +--R (1) cosh(-) == -------------- +--R 2 +-+ +--R \|2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 94 + +--S 95 of 298 +t1:=cosh(x/2) - sqrt((cosh(x)+1)/2) +--R +--R +-----------+ +-+ x +--R - \|cosh(x) + 1 + \|2 cosh(-) +--R 2 +--R (2) ------------------------------ +--R +-+ +--R \|2 +--R Type: Expression Integer +--E 95 + +--S 96 of 298 +t2:=coshhalf t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 96 + +@ +\subsection{8.29 ${\rm tanh}\left(\frac{x}{2}\right)$} +\[{\rm tanh}\left(\frac{x}{2}\right) == \pm\sqrt{\frac{{\rm cosh}(x) - 1}{{\rm cosh}(x) + 1}}\quad [+~{\rm if\ \ }x > 0, -{\rm if\ \ }x < 0]\] +\[== \frac{{\rm sinh}(x)}{{\rm cosh}(x)+1}\] +\[== \frac{{\rm cosh}(x)-1}{{\rm sinh}(x)}\] + +if $x > 0$ +<<*>>= +)clear all + +--S 97 of 298 +tanhhalf:=rule(tanh(x/2) == sqrt((cosh(x)-1)/(cosh(x)+1))) +--R +--R +-----------+ +--R x |cosh(x) - 1 +--R (1) tanh(-) == |----------- +--R 2 \|cosh(x) + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 97 + +--S 98 of 298 +t1:=tanh(x/2) -sqrt((cosh(x)-1)/(cosh(x)+1)) +--R +--R +-----------+ +--R |cosh(x) - 1 x +--R (2) - |----------- + tanh(-) +--R \|cosh(x) + 1 2 +--R Type: Expression Integer +--E 98 + +--S 99 of 298 +t2:=tanhhalf t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 99 + +@ +if $x < 0$ +<<*>>= +)clear all + +--S 100 of 298 +tanhhalfneg:=rule(tanh(x/2) == -sqrt((cosh(x)-1)/(cosh(x)+1))) +--R +--R +-----------+ +--R x |cosh(x) - 1 +--R (1) tanh(-) == - |----------- +--R 2 \|cosh(x) + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 100 + +--S 101 of 298 +t1:=tanh(x/2) - -sqrt((cosh(x)-1)/(cosh(x)+1)) +--R +--R +-----------+ +--R |cosh(x) - 1 x +--R (2) |----------- + tanh(-) +--R \|cosh(x) + 1 2 +--R Type: Expression Integer +--E 101 + +--S 102 of 298 +t2:=tanhhalfneg t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 102 + +)clear all + +--S 103 of 298 +tanhhalf2:=rule(tanh(x/2) == sinh(x)/(cosh(x)+1)) +--R +--R x sinh(x) +--R (1) tanh(-) == ----------- +--R 2 cosh(x) + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 103 + +--S 104 of 298 +t1:=tanh(x/2) - sinh(x)/(cosh(x)+1) +--R +--R x +--R (cosh(x) + 1)tanh(-) - sinh(x) +--R 2 +--R (2) ------------------------------ +--R cosh(x) + 1 +--R Type: Expression Integer +--E 104 + +--S 105 of 298 +t2:=tanhhalf2 t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 105 + +)clear all + +--S 106 of 298 +tanhhalf3:=rule(tanh(x/2) == (cosh(x)-1)/sinh(x)) +--R +--R x cosh(x) - 1 +--R (1) tanh(-) == ----------- +--R 2 sinh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 106 + +--S 107 of 298 +t1:=tanh(x/2) - (cosh(x)-1)/sinh(x) +--R +--R x +--R sinh(x)tanh(-) - cosh(x) + 1 +--R 2 +--R (2) ---------------------------- +--R sinh(x) +--R Type: Expression Integer +--E 107 + +--S 108 of 298 +t2:=tanhhalf3 t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 108 + +@ + +\section{Multiple Angle Formulas} +\subsection{8.30 ${\rm sinh}(3x)$} +\[{\rm sinh}(3x) == 3~{\rm sinh}(x)+4~{\rm sinh}^3(x)\] +<<*>>= +)clear all + +--S 109 of 298 +sinh3x:=rule(sinh(3*x) == 3*sinh(x)+4*sinh(x)^3) +--R +--R 3 +--R (1) sinh(3x) == 4sinh(x) + 3sinh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 109 + +--S 110 of 298 +t1:=sinh(3*x) - (3*sinh(x)+4*sinh(x)^3) +--R +--R 3 +--R (2) sinh(3x) - 4sinh(x) - 3sinh(x) +--R Type: Expression Integer +--E 110 + +--S 111 of 298 +t2:=sinh3x t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 111 + +@ +\subsection{8.31 ${\rm cosh}(3x)$} +\[{\rm cosh}(3x) == 4~{\rm cosh}^3(x)-3~{\rm cosh}(x)\] +<<*>>= +)clear all + +--S 112 of 298 +cosh3x:=rule(cosh(3*x) == 4*cosh(x)^3-3*cosh(x)) +--R +--R 3 +--R (1) cosh(3x) == 4cosh(x) - 3cosh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 112 + +--S 113 of 298 +t1:=cosh(3*x) - (4*cosh(x)^3-3*cosh(x)) +--R +--R 3 +--R (2) cosh(3x) - 4cosh(x) + 3cosh(x) +--R Type: Expression Integer +--E 113 + +--S 114 of 298 +t2:=cosh3x t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 114 + +@ +\subsection{8.32 ${\rm tanh}(3x)$} +\[{\rm tanh}(3x) == \frac{3~{\rm tanh}(x)+{\rm tanh}^3(x)}{1+3~{\rm tanh}^2(x)}\] +<<*>>= +)clear all + +--S 115 of 298 +tanh3x:=rule(tanh(3*x) == (3*tanh(x)+tanh(x)^3)/(1+3*tanh(x)^2)) +--R +--R 3 +--R tanh(x) + 3tanh(x) +--R (1) tanh(3x) == ------------------- +--R 2 +--R 3tanh(x) + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 115 + +--S 116 of 298 +t1:=tanh(3*x) - (3*tanh(x)+tanh(x)^3)/(1+3*tanh(x)^2) +--R +--R 2 3 +--R (3tanh(x) + 1)tanh(3x) - tanh(x) - 3tanh(x) +--R (2) --------------------------------------------- +--R 2 +--R 3tanh(x) + 1 +--R Type: Expression Integer +--E 116 + +--S 117 of 298 +t2:=tanh3x t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 117 + +@ +\subsection{8.33 ${\rm sinh}(4x)$} +\[{\rm sinh}(4x) == 8~{\rm sinh}^3(x)~{\rm cosh}(x)+4~{\rm sinh}(x)~{\rm cosh}(x)\] +<<*>>= +)clear all + +--S 118 of 298 +sinh4x:=rule(sinh(4*x) == 8*sinh(x)^3*cosh(x)+4*sinh(x)*cosh(x)) +--R +--R 3 +--R (1) sinh(4x) == 8cosh(x)sinh(x) + 4cosh(x)sinh(x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 118 + +--S 119 of 298 +t1:=sinh(4*x) - (8*sinh(x)^3*cosh(x)+4*sinh(x)*cosh(x)) +--R +--R 3 +--R (2) sinh(4x) - 8cosh(x)sinh(x) - 4cosh(x)sinh(x) +--R Type: Expression Integer +--E 119 + +--S 120 of 298 +t2:=sinh4x t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 120 + +@ +\subsection{8.34 ${\rm cosh}(4x)$} +\[{\rm cosh}(4x) == 8~{\rm cosh}^4(x)-8~{\rm cosh}^2(x)+1\] +<<*>>= +)clear all + +--S 121 of 298 +cosh4x:=rule(cosh(4*x) == 8*cosh(x)^4-8*cosh(x)^2+1) +--R +--R 4 2 +--R (1) cosh(4x) == 8cosh(x) - 8cosh(x) + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 121 + +--S 122 of 298 +t1:=cosh(4*x) - (8*cosh(x)^4-8*cosh(x)^2+1) +--R +--R 4 2 +--R (2) cosh(4x) - 8cosh(x) + 8cosh(x) - 1 +--R Type: Expression Integer +--E 122 + +--S 123 of 298 +t2:=cosh4x t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 123 + +@ +\subsection{8.35 ${\rm tanh}(4x)$} +\[{\rm tanh}(4x) == \frac{4~{\rm tanh}(x)+4~{\rm tanh}^3(x)}{1+6~{\rm tanh}^2(x)+{\rm tanh}^4(x)}\] +<<*>>= +)clear all + +--S 124 of 298 +tanh4x:=rule(tanh(4*x) == (4*tanh(x)+4*tanh(x)^3)/(1+6*tanh(x)^2+tanh(x)^4)) +--R +--R 3 +--R 4tanh(x) + 4tanh(x) +--R (1) tanh(4x) == ------------------------ +--R 4 2 +--R tanh(x) + 6tanh(x) + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 124 + +--S 125 of 298 +t1:=tanh(4*x) - (4*tanh(x)+4*tanh(x)^3)/(1+6*tanh(x)^2+tanh(x)^4) +--R +--R 4 2 3 +--R (tanh(x) + 6tanh(x) + 1)tanh(4x) - 4tanh(x) - 4tanh(x) +--R (2) --------------------------------------------------------- +--R 4 2 +--R tanh(x) + 6tanh(x) + 1 +--R Type: Expression Integer +--E 125 + +--S 126 of 298 +t2:=tanh4x t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 126 + +@ + +\section{Powers of Hyperbolic Functions} +\subsection{8.36 ${\rm sinh}^2(x)$} +\[{\rm sinh}^2(x) == \frac{1}{2}~{\rm cosh}(2x)-\frac{1}{2}\] +<<*>>= +)clear all + +--S 127 of 298 +sinhsquared:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2) +--R +--R 2 cosh(2x) - 1 +--R (1) sinh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 127 + +--S 128 of 298 +t1:=sinh(x)^2 - (1/2*cosh(2*x)-1/2) +--R +--R 2 +--R 2sinh(x) - cosh(2x) + 1 +--R (2) ------------------------ +--R 2 +--R Type: Expression Integer +--E 128 + +--S 129 of 298 +t2:=sinhsquared t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 129 + +@ +\subsection{8.37 ${\rm cosh}^2(x)$} +\[{\rm cosh}^2(x) == \frac{1}{2}~{\rm cosh}(2x)+\frac{1}{2}\] +<<*>>= +)clear all + +--S 130 of 298 +coshsquared:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2) +--E 130 + +--S 131 of 298 +t1:=cosh(x)^2 - (1/2*cosh(2*x)+1/2) +--E 131 + +--S 132 of 298 +t2:=coshsquared t1 +--E 132 + +@ +\subsection{8.38 ${\rm sinh}^3(x)$} +\[{\rm sinh}^3(x) == \frac{1}{4}~{\rm sinh}(3x)-\frac{3}{4}~{\rm sinh}(x)\] +<<*>>= +)clear all + +--S 133 of 298 +sinhcubed:=rule(sinh(x)^3 == 1/4*sinh(3*x)-3/4*sinh(x)) +--R +--R 3 sinh(3x) - 3sinh(x) +--R (1) sinh(x) == ------------------- +--R 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 133 + +--S 134 of 298 +t1:=sinh(x)^3 - (1/4*sinh(3*x)-3/4*sinh(x)) +--R +--R 3 +--R - sinh(3x) + 4sinh(x) + 3sinh(x) +--R (2) --------------------------------- +--R 4 +--R Type: Expression Integer +--E 134 + +--S 135 of 298 +t2:=sinhcubed t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 135 + +@ +\subsection{8.39 ${\rm cosh}^3(x)$} +\[{\rm cosh}^3(x) == \frac{1}{4}~{\rm cosh}(3x)+\frac{3}{4}~{\rm cosh}(x)\] +<<*>>= +)clear all + +--S 136 of 298 +coshcubed:=rule(cosh(x)^3 == 1/4*cosh(3*x)+3/4*cosh(x)) +--R +--R 3 cosh(3x) + 3cosh(x) +--R (1) cosh(x) == ------------------- +--R 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 136 + +--S 137 of 298 +t1:=cosh(x)^3 - (1/4*cosh(3*x)+3/4*cosh(x)) +--R +--R 3 +--R - cosh(3x) + 4cosh(x) - 3cosh(x) +--R (2) --------------------------------- +--R 4 +--R Type: Expression Integer +--E 137 + +--S 138 of 298 +t2:=coshcubed t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 138 + +@ +\subsection{8.40 ${\rm sinh}^4(x)$} +\[{\rm sinh}^4(x) == \frac{3}{8}-\frac{1}{2}~{\rm cosh}(2x)+\frac{1}{8}~{\rm cosh}(4x)\] +<<*>>= +)clear all + +--S 139 of 298 +sinhfourth:=rule(sinh(x)^4 == 3/8-1/2*cosh(2*x)+1/8*cosh(4*x)) +--R +--R 4 cosh(4x) - 4cosh(2x) + 3 +--R (1) sinh(x) == ------------------------ +--R 8 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 139 + +--S 140 of 298 +t1:=sinh(x)^4 - (3/8-1/2*cosh(2*x)+1/8*cosh(4*x)) +--R +--R 4 +--R 8sinh(x) - cosh(4x) + 4cosh(2x) - 3 +--R (2) ------------------------------------ +--R 8 +--R Type: Expression Integer +--E 140 + +--S 141 of 298 +t2:=sinhfourth t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 141 + +@ +\subsection{8.41 ${\rm cosh}^4(x)$} +\[{\rm cosh}^4(x) == \frac{3}{8}+\frac{1}{2}~{\rm cosh}(2x)+\frac{1}{8}~{\rm cosh}(4x)\] +<<*>>= +)clear all + +--S 142 of 298 +coshfourth:=rule(cosh(x)^4 == 3/8+1/2*cosh(2*x)+1/8*cosh(4*x)) +--R +--R 4 cosh(4x) + 4cosh(2x) + 3 +--R (1) cosh(x) == ------------------------ +--R 8 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 142 + +--S 143 of 298 +t1:=cosh(x)^4 - (3/8+1/2*cosh(2*x)+1/8*cosh(4*x)) +--R +--R 4 +--R - cosh(4x) - 4cosh(2x) + 8cosh(x) - 3 +--R (2) -------------------------------------- +--R 8 +--R Type: Expression Integer +--E 143 + +--S 144 of 298 +t2:=coshfourth t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 144 + +@ +\section{Sum, Difference, and Product of Hyperbolic Products} +\subsection{8.42 ${\rm sinh}(x)+{\rm sinh}(y)$} +\[{\rm sinh}(x)+{\rm sinh}(y) == 2~{\rm sinh}(\frac{1}{2}(x+y))~{\rm cosh}(\frac{1}{2}(x-y))\] +<<*>>= +)clear all + +--S 145 of 298 +sinhplussinh:=rule(sinh(x)+sinh(y) == 2*sinh(1/2*(x+y))*cosh(1/2*(x-y))) +--R +--R y - x y + x +--I (1) sinh(y) + sinh(x) + %M == 2cosh(-----)sinh(-----) + %M +--R 2 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 145 + +--S 146 of 298 +t1:=sinh(x)+sinh(y) - 2*sinh(1/2*(x+y))*cosh(1/2*(x-y)) +--R +--R y - x y + x +--R (2) sinh(y) - 2cosh(-----)sinh(-----) + sinh(x) +--R 2 2 +--R Type: Expression Integer +--E 146 + +--S 147 of 298 +t2:=sinhplussinh t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 147 + +@ +\subsection{8.43 ${\rm sinh}(x)-{\rm sinh}(y)$} +\[{\rm sinh}(x)-{\rm sinh}(y) == 2~{\rm cosh}(\frac{1}{2}(x+y))~{\rm sinh}(\frac{1}{2}(x-y))\] +<<*>>= +)clear all + +--S 148 of 298 +sinhminussinh:=rule(sinh(x)-sinh(y) == 2*cosh(1/2*(x+y))*sinh(1/2*(x-y))) +--R +--R y + x y - x +--I (1) - sinh(y) + sinh(x) + %N == - 2cosh(-----)sinh(-----) + %N +--R 2 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 148 + +--S 149 of 298 +t1:=sinh(x)-sinh(y) - 2*cosh(1/2*(x+y))*sinh(1/2*(x-y)) +--R +--R y + x y - x +--R (2) - sinh(y) + 2cosh(-----)sinh(-----) + sinh(x) +--R 2 2 +--R Type: Expression Integer +--E 149 + +--S 150 of 298 +t2:=sinhminussinh t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 150 + +@ +\subsection{8.44 ${\rm cosh}(x)+{\rm cosh}(y)$} +\[{\rm cosh}(x)+{\rm cosh}(y) == 2~{\rm cosh}(\frac{1}{2}(x+y))~{\rm cosh}(\frac{1}{2}(x-y))\] +<<*>>= +)clear all + +--S 151 of 298 +coshpluscosh:=rule(cosh(x)+cosh(y) == 2*cosh(1/2*(x+y))*cosh(1/2*(x-y))) +--R +--R y - x y + x +--I (1) cosh(y) + cosh(x) + %O == 2cosh(-----)cosh(-----) + %O +--R 2 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 151 + +--S 152 of 298 +t1:=cosh(x)+cosh(y) - 2*cosh(1/2*(x+y))*cosh(1/2*(x-y)) +--R +--R y - x y + x +--R (2) cosh(y) - 2cosh(-----)cosh(-----) + cosh(x) +--R 2 2 +--R Type: Expression Integer +--E 152 + +--S 153 of 298 +t2:=coshpluscosh t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 153 + +@ +\subsection{8.45 ${\rm cosh}(x)-{\rm cosh}(y)$} +\[{\rm cosh}(x)-{\rm cosh}(y) == 2~{\rm sinh}(\frac{1}{2}(x+y))~{\rm sinh}(\frac{1}{2}(x-y))\] +<<*>>= +)clear all + +--S 154 of 298 +coshminuscosh:=rule(cosh(x)-cosh(y) == 2*sinh(1/2*(x+y))*sinh(1/2*(x-y))) +--R +--R y - x y + x +--I (1) - cosh(y) + cosh(x) + %P == - 2sinh(-----)sinh(-----) + %P +--R 2 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 154 + +--S 155 of 298 +t1:=cosh(x)-cosh(y) - 2*sinh(1/2*(x+y))*sinh(1/2*(x-y)) +--R +--R y - x y + x +--R (2) 2sinh(-----)sinh(-----) - cosh(y) + cosh(x) +--R 2 2 +--R Type: Expression Integer +--E 155 + +--S 156 of 298 +t2:=coshminuscosh t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 156 + +@ +\subsection{8.46 ${\rm sinh}(x){\rm sinh}(y)$} +\[{\rm sinh}(x){\rm sinh}(y) == \frac{1}{2}(~{\rm cosh}(x+y)-{\rm cosh}(x-y))\] +<<*>>= +)clear all + +--S 157 of 298 +sinhtimessinh:=rule(sinh(x)*sinh(y) == 1/2*(cosh(x+y)-cosh(x-y))) +--R +--I %Q cosh(y + x) - %Q cosh(y - x) +--I (1) %Q sinh(x)sinh(y) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 157 + +--S 158 of 298 +t1:=sinh(x)*sinh(y) - (1/2*(cosh(x+y)-cosh(x-y))) +--R +--R 2sinh(x)sinh(y) - cosh(y + x) + cosh(y - x) +--R (2) ------------------------------------------- +--R 2 +--R Type: Expression Integer +--E 158 + +--S 159 of 298 +t2:=sinhtimessinh t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 159 + +@ +\subsection{8.47 ${\rm cosh}(x){\rm cosh}(y)$} +\[{\rm cosh}(x){\rm cosh}(y) == \frac{1}{2}(~{\rm cosh}(x+y)+{\rm cosh}(x-y))\] +<<*>>= +)clear all + +--S 160 of 298 +coshtimescosh:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y))) +--R +--I %R cosh(y + x) + %R cosh(y - x) +--I (1) %R cosh(x)cosh(y) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 160 + +--S 161 of 298 +t1:=cosh(x)*cosh(y) - 1/2*(cosh(x+y)+cosh(x-y)) +--R +--R - cosh(y + x) + 2cosh(x)cosh(y) - cosh(y - x) +--R (2) --------------------------------------------- +--R 2 +--R Type: Expression Integer +--E 161 + +--S 162 of 298 +t2:=coshtimescosh t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 162 + +@ +\subsection{8.48 ${\rm sinh}(x){\rm cosh}(y)$} +\[{\rm sinh}(x){\rm cosh}(y) == \frac{1}{2}(~{\rm sinh}(x+y)+{\rm sinh}(x-y))\] +<<*>>= +)clear all + +--S 163 of 298 +sinhtimescosh:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y))) +--R +--I %S sinh(y + x) - %S sinh(y - x) +--I (1) %S cosh(y)sinh(x) == ------------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 163 + +--S 164 of 298 +t1:=sinh(x)*cosh(y) - 1/2*(sinh(x+y)+sinh(x-y)) +--R +--R - sinh(y + x) + sinh(y - x) + 2cosh(y)sinh(x) +--R (2) --------------------------------------------- +--R 2 +--R Type: Expression Integer +--E 164 + +--S 165 of 298 +t2:=sinhtimescosh t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 165 + +@ +\section{Inverse Hyperbolic Functions} +\subsection{8.55 ${\rm sinh}^{-1}(x)$} +\[{\rm asinh}(x) == \log(x+\sqrt{x^2+1}) \quad {\rm if\ \ } -\infty < x < \infty\] +<<*>>= +)clear all + +--S 166 of 298 +asinhrule:=rule(asinh(x) == log(x+sqrt(x^2+1))) +--R +--R +------+ +--R | 2 +--R (1) asinh(x) == log(\|x + 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 166 + +--S 167 of 298 +t1:=asinh(x) - log(x+sqrt(x^2+1)) +--R +--R +------+ +--R | 2 +--R (2) - log(\|x + 1 + x) + asinh(x) +--R Type: Expression Integer +--E 167 + +--S 168 of 298 +t2:=asinhrule t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 168 + +@ +\subsection{8.56 ${\rm cosh}^{-1}(x)$} +\[{\rm acosh}(x) == \log(x+\sqrt{x^2-1}) \quad {\rm if\ \ } x \ge 1\] +<<*>>= +)clear all + +--S 169 of 298 +acoshrule:=rule(acosh(x) == log(x+sqrt(x^2-1))) +--R +--R +------+ +--R | 2 +--R (1) acosh(x) == log(\|x - 1 + x) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 169 + +--S 170 of 298 +t1:=acosh(x) - log(x+sqrt(x^2-1)) +--R +--R +------+ +--R | 2 +--R (2) - log(\|x - 1 + x) + acosh(x) +--R Type: Expression Integer +--E 170 + +--S 171 of 298 +t2:=acoshrule t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 171 + +@ + +\subsection{8.57 ${\rm tanh}^{-1}(x)$} +\[{\rm atanh}(x) == \frac{1}{2}\log(\frac{1+x}{1-x}) \quad {\rm if\ \ } -1 < x < 1\] +<<*>>= +)clear all + +--S 172 of 298 +atanhrule:=rule(atanh(x) == 1/2*log((1+x)/(1-x))) +--R +--R - x - 1 +--R log(-------) +--R x - 1 +--R (1) atanh(x) == ------------ +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 172 + +--S 173 of 298 +t1:=atanh(x) - 1/2*log((1+x)/(1-x)) +--R +--R - x - 1 +--R - log(-------) + 2atanh(x) +--R x - 1 +--R (2) -------------------------- +--R 2 +--R Type: Expression Integer +--E 173 + +--S 174 of 298 +t2:=atanhrule t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 174 + +@ + +\subsection{8.58 ${\rm coth}^{-1}(x)$} +\[{\rm acoth}(x) == \frac{1}{2}\log(\frac{x+1}{x-1}) \quad {\rm if\ \ } x>1 or x<-1\] +<<*>>= +)clear all + +--S 175 of 298 +acothrule:=rule(acoth(x) == 1/2*log((x+1)/(x-1))) +--R +--R x + 1 +--R log(-----) +--R x - 1 +--R (1) acoth(x) == ---------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 175 + +--S 176 of 298 +t1:=acoth(x) - 1/2*log((x+1)/(x-1)) +--R +--R x + 1 +--R - log(-----) + 2acoth(x) +--R x - 1 +--R (2) ------------------------ +--R 2 +--R Type: Expression Integer +--E 176 + +--S 177 of 298 +t2:=acothrule t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 177 + +@ + +\subsection{8.59 ${\rm sech}^{-1}(x)$} +\[{\rm asech}(x) == \log(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}) \quad {\rm if\ \ } 0 < x \le 1\] +<<*>>= +)clear all + +--S 178 of 298 +asechrule:=rule(asech(x) == log(1/x+sqrt(1/x^2-1))) +--R +--R +--------+ +--R | 2 +--R |- x + 1 +--R x |-------- + 1 +--R | 2 +--R \| x +--R (1) asech(x) == log(----------------) +--R x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 178 + +--S 179 of 298 +t1:=asech(x) - log(1/x+sqrt(1/x^2-1)) +--R +--R +--------+ +--R | 2 +--R |- x + 1 +--R x |-------- + 1 +--R | 2 +--R \| x +--R (2) - log(----------------) + asech(x) +--R x +--R Type: Expression Integer +--E 179 + +--S 180 of 298 +t2:=asechrule t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 180 + +@ +\subsection{8.60 ${\rm csch}^{-1}(x)$} +\[{\rm acsch}(x) == \log(\frac{1}{x}+\sqrt{\frac{1}{x^2}+1}) \quad {\rm if\ \ } x \ne 0\] +<<*>>= +)clear all + +--S 181 of 298 +acschrule:=rule(acsch(x) == log(1/x+sqrt(1/x^2+1))) +--R +--R +------+ +--R | 2 +--R |x + 1 +--R x |------ + 1 +--R | 2 +--R \| x +--R (1) acsch(x) == log(--------------) +--R x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 181 + +--S 182 of 298 +t1:=acsch(x) - log(1/x+sqrt(1/x^2+1)) +--R +--R +------+ +--R | 2 +--R |x + 1 +--R x |------ + 1 +--R | 2 +--R \| x +--R (2) - log(--------------) + acsch(x) +--R x +--R Type: Expression Integer +--E 182 + +--S 183 of 298 +t2:=acschrule t1 +--R +--R (3) 0 +--R Type: Expression Integer +--E 183 + +@ +\section{Relations between Inverse Hyperbolic Functions} +\subsection{8.61 ${\rm csch}^{-1}(x)$} +\[{\rm csch}^{-1}(x) == {\rm sinh}^{-1}\left(\frac{1}{x}\right)\] +<<*>>= +)clear all + +--S 184 of 298 +cschinv:=rule(csch(x)^(-1) == sinh(1/x)^(-1)) +--R +--R 1 1 +--R (1) ------- == ------- +--R csch(x) 1 +--R sinh(-) +--R x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 184 + +--S 185 of 298 +t1:=csch(x)^(-1) - sinh(1/x)^(-1) +--R +--R 1 +--R sinh(-) - csch(x) +--R x +--R (2) ----------------- +--R 1 +--R csch(x)sinh(-) +--R x +--R Type: Expression Integer +--E 185 + +--S 186 of 298 +t2:=cschinv t1 +--R +--R 1 +--R sinh(-) - csch(x) +--R x +--R (3) ----------------- +--R 1 +--R csch(x)sinh(-) +--R x +--R Type: Expression Integer +--E 186 + +@ +\subsection{8.62 ${\rm sech}^{-1}(x)$} +\[{\rm sech}^{-1}(x) == {\rm cosh}^{-1}\left(\frac{1}{x}\right)\] +<<*>>= +)clear all + +--S 187 of 298 +sechinv:=rule(sech(x)^(-1) == cosh(1/x)^(-1)) +--R +--R 1 1 +--R (1) ------- == ------- +--R sech(x) 1 +--R cosh(-) +--R x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 187 + +--S 188 of 298 +t1:=sech(x)^(-1) - cosh(1/x)^(-1) +--R +--R 1 +--R - sech(x) + cosh(-) +--R x +--R (2) ------------------- +--R 1 +--R cosh(-)sech(x) +--R x +--R Type: Expression Integer +--E 188 + +--S 189 of 298 +t2:=sechinv t1 +--R +--R 1 +--R - sech(x) + cosh(-) +--R x +--R (3) ------------------- +--R 1 +--R cosh(-)sech(x) +--R x +--R Type: Expression Integer +--E 189 + +@ +\subsection{8.63 ${\rm coth}^{-1}(x)$} +\[{\rm coth}^{-1}(x) == {\rm tanh}^{-1}\left(\frac{1}{x}\right)\] +<<*>>= +)clear all + +--S 190 of 298 +cothinv:=rule(coth(x)^(-1) == tanh(1/x)^(-1)) +--R +--R 1 1 +--R (1) ------- == ------- +--R coth(x) 1 +--R tanh(-) +--R x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E 190 + +--S 191 of 298 +t1:=coth(x)^(-1) - tanh(1/x)^(-1) +--R +--R 1 +--R tanh(-) - coth(x) +--R x +--R (2) ----------------- +--R 1 +--R coth(x)tanh(-) +--R x +--R Type: Expression Integer +--E 191 + +--S 192 of 298 +t2:=cothinv t1 +--R +--R 1 +--R tanh(-) - coth(x) +--R x +--R (3) ----------------- +--R 1 +--R coth(x)tanh(-) +--R x +--R Type: Expression Integer +--E 192 + +@ +\subsection{8.64 ${\rm sinh}^{-1}(x)$} +\[{\rm sinh}^{-1}(x) == -~{\rm sinh}^{-1}(x)\] +These identities are already known to Axiom +<<*>>= +)clear all + +--S 193 of 298 +t1:=sinh(-x)^(-1) - -sinh(x)^(-1) +--R +--R (1) 0 +--R Type: Expression Integer +--E 193 + +@ +\subsection{8.65 ${\rm tanh}^{-1}(x)$} +\[{\rm tanh}^{-1}(x) == -~{\rm tanh}^{-1}(x)\] +<<*>>= +)clear all + +--S 194 of 298 +t1:=tanh(-x)^(-1) - -tanh(x)^(-1) +--R +--R (1) 0 +--R Type: Expression Integer +--E 194 + +@ +\subsection{8.66 ${\rm coth}^{-1}(x)$} +\[{\rm coth}^{-1}(x) == -~{\rm coth}^{-1}(x)\] +<<*>>= +)clear all + +--S 195 of 298 +t1:=coth(-x)^(-1) - -coth(x)^(-1) +--R +--R (1) 0 +--R Type: Expression Integer +--E 195 + +@ +\subsection{8.67 ${\rm csch}^{-1}(x)$} +\[{\rm csch}^{-1}(x) == -~{\rm csch}^{-1}(x)\] +<<*>>= +)clear all + +--S 196 of 298 +t1:=csch(-x)^(-1) - -csch(x)^(-1) +--R +--R (1) 0 +--R Type: Expression Integer +--E 196 + +@ +\section{Relationship between Hyperbolic and Trigonometric Functions} +\subsection{8.74 $\sin(ix)$} +\[\sin(ix) == i~{\rm sinh}(x)\] +This match does not work. +<<*>>= +)clear all + +--S 197 of 298 +sininv:=rule(sin(%i*x) == %i*sinh(x)) +--R +--R (1) sin(%i x) == %i sinh(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 197 + +--S 198 of 298 +t1:=sin(x*%i) - %i*sinh(x) +--R +--R (2) - %i sinh(x) + sin(%i x) +--R Type: Expression Complex Integer +--E 198 + +--S 199 of 298 +t2:=sininv t1 +--R +--R (3) - %i sinh(x) + sin(%i x) +--R Type: Expression Complex Integer +--E 199 + +@ +\subsection{8.75 $\cos(ix)$} +\[\cos(ix) == {\rm cosh}(x)\] +<<*>>= +)clear all + +--S 200 of 298 +cosinv:=rule(cos(x*%i) == cosh(x)) +--R +--R (1) cos(%i x) == cosh(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 200 + +--S 201 of 298 +t1:=cos(x*%i) - cosh(x) +--R +--R (2) - cosh(x) + cos(%i x) +--R Type: Expression Complex Integer +--E 201 + +--S 202 of 298 +t2:=cosinv t1 +--R +--R (3) - cosh(x) + cos(%i x) +--R Type: Expression Complex Integer +--E 202 + +@ +\subsection{8.76 $\tan(ix)$} +\[\tan(ix) == i~{\rm tanh}(x)\] +<<*>>= +)clear all + +--S 203 of 298 +taninv:=rule(tan(x*%i) == %i*tanh(x)) +--R +--R (1) tan(%i x) == %i tanh(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 203 + +--S 204 of 298 +t1:=tan(x*%i) - %i*tanh(x) +--R +--R (2) - %i tanh(x) + tan(%i x) +--R Type: Expression Complex Integer +--E 204 + +--S 205 of 298 +t2:=taninv t1 +--R +--R (3) - %i tanh(x) + tan(%i x) +--R Type: Expression Complex Integer +--E 205 + +@ +\subsection{8.77 $\csc(ix)$} +\[\csc(ix) == -i~{\rm csch}(x)\] +<<*>>= +)clear all + +--S 206 of 298 +cscinv:=rule(csc(x*%i) == -%i*csch(x)) +--R +--R (1) csc(%i x) == - %i csch(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 206 + +--S 207 of 298 +t1:=csc(x*%i) - -%i*csch(x) +--R +--R (2) %i csch(x) + csc(%i x) +--R Type: Expression Complex Integer +--E 207 + +--S 208 of 298 +t2:=cscinv t1 +--R +--R (3) %i csch(x) + csc(%i x) +--R Type: Expression Complex Integer +--E 208 + +@ +\subsection{8.78 $\sec(ix)$} +\[\sec(ix) == {\rm sech}(x)\] +<<*>>= +)clear all + +--S 209 of 298 +secinv:=rule(sec(x*%i) == sech(x)) +--R +--R (1) sec(%i x) == sech(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 209 + +--S 210 of 298 +t1:=sec(x*%i) - sech(x) +--R +--R (2) - sech(x) + sec(%i x) +--R Type: Expression Complex Integer +--E 210 + +--S 211 of 298 +t2:=secinv t1 +--R +--R (3) - sech(x) + sec(%i x) +--R Type: Expression Complex Integer +--E 211 + +@ +\subsection{8.79 $\cot(ix)$} +\[\cot(ix) == -i~{\rm coth}(x)\] +<<*>>= +)clear all + +--S 212 of 298 +cotinv:=rule(cot(x*%i) == -%i*coth(x)) +--R +--R (1) cot(%i x) == - %i coth(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 212 + +--S 213 of 298 +t1:=cot(x*%i) - -%i*coth(x) +--R +--R (2) %i coth(x) + cot(%i x) +--R Type: Expression Complex Integer +--E 213 + +--S 214 of 298 +t2:=cotinv t1 +--R +--R (3) %i coth(x) + cot(%i x) +--R Type: Expression Complex Integer +--E 214 + +@ +\subsection{8.80 ${\rm sinh}(ix)$} +\[\sinh(ix) == i~\sin(x)\] +<<*>>= +)clear all + +--S 215 of 298 +sinhinv:=rule(sinh(x*%i) == %i*sin(x)) +--R +--R (1) sinh(%i x) == %i sin(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 215 + +--S 216 of 298 +t1:=sinh(x*%i) - %i*sin(x) +--R +--R (2) sinh(%i x) - %i sin(x) +--R Type: Expression Complex Integer +--E 216 + +--S 217 of 298 +t2:=sinhinv t1 +--R +--R (3) sinh(%i x) - %i sin(x) +--R Type: Expression Complex Integer +--E 217 + +@ +\subsection{8.81 ${\rm cosh}(ix)$} +\[\cosh(ix) == \cos(x)\] +<<*>>= +)clear all + +--S 218 of 298 +coshinv:=rule(cosh(x*%i) == cos(x)) +--R +--R (1) cosh(%i x) == cos(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 218 + +--S 219 of 298 +t1:=cosh(x*%i) - cos(x) +--R +--R (2) cosh(%i x) - cos(x) +--R Type: Expression Complex Integer +--E 219 + +--S 220 of 298 +t2:=coshinv t1 +--R +--R (3) cosh(%i x) - cos(x) +--R Type: Expression Complex Integer +--E 220 + +@ +p\subsection{8.82 ${\rm tanh}(ix)$} +\[\tanh(ix) == i\tan(x)\] +<<*>>= +)clear all + +--S 221 of 298 +tanhinv:=rule(tanh(x*%i) == %i*tan(x)) +--R +--R (1) tanh(%i x) == %i tan(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 221 + +--S 222 of 298 +t1:=tanh(x*%i) - %i*tan(x) +--R +--R (2) tanh(%i x) - %i tan(x) +--R Type: Expression Complex Integer +--E 222 + +--S 223 of 298 +t2:=tanhinv t1 +--R +--R (3) tanh(%i x) - %i tan(x) +--R Type: Expression Complex Integer +--E 223 + +@ +\subsection{8.83 ${\rm csch}(ix)$} +\[{\rm csch}(ix) == -i \csc(x)\] +<<*>>= +)clear all + +--S 224 of 298 +cschinv:=rule(x*%i == -%i*csc(x)) +--R +--R (1) %i x == - %i csc(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 224 + +--S 225 of 298 +t1:=x*%i - -%i*csc(x) +--R +--R (2) %i csc(x) + %i x +--R Type: Expression Complex Integer +--E 225 + +--S 226 of 298 +t2:=cschinv t1 +--R +--R (3) %i csc(x) + %i x +--R Type: Expression Complex Integer +--E 226 + +@ +\subsection{8.84 ${\rm sech}(ix)$} +\[{\rm sech}(ix) == \sec(x)\] +<<*>>= +)clear all + +--S 227 of 298 +sechinv:=rule(sech(x*%i) == sec(x)) +--R +--R (1) sech(%i x) == sec(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 227 + +--S 228 of 298 +t1:=sech(x*%i) - sec(x) +--R +--R (2) sech(%i x) - sec(x) +--R Type: Expression Complex Integer +--E 228 + +--S 229 of 298 +t2:=sechinv t1 +--R +--R (3) sech(%i x) - sec(x) +--R Type: Expression Complex Integer +--E 229 + +@ +\subsection{8.85 ${\rm coth}(ix)$} +\[{\rm coth}(ix) == -i \cot(x)\] +<<*>>= +)clear all + +--S 230 of 298 +cothinv:=rule(coth(x*%i) == -%i*cot(x)) +--R +--R (1) coth(%i x) == - %i cot(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 230 + +--S 231 of 298 +t1:=coth(x*%i) - -%i*cot(x) +--R +--R (2) coth(%i x) + %i cot(x) +--R Type: Expression Complex Integer +--E 231 + +--S 232 of 298 +t2:=cothinv t1 +--R +--R (3) coth(%i x) + %i cot(x) +--R Type: Expression Complex Integer +--E 232 + +@ + +\section{Periodicity of Hyperbolic Functions} +\subsection{8.86 ${\rm sinh}(x+2k\pi i)$} +\[{\rm sinh}(x+2k\pi i) == {\rm sinh}(x)\] +<<*>>= +)clear all + +--S 233 of 298 +sinhperiod:=rule(sinh(x+2*k*%pi*%i) == sinh(x)) +--R +--R (1) sinh(x + 2%i k %pi) == sinh(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 233 + +--S 234 of 298 +t1:=sinh(x+2*k*%pi*%i) - sinh(x) +--R +--R (2) sinh(x + 2%i k %pi) - sinh(x) +--R Type: Expression Complex Integer +--E 234 + +--S 235 of 298 +t2:=sinhperiod t1 +--R +--R (3) sinh(x + 2%i k %pi) - sinh(x) +--R Type: Expression Complex Integer +--E 235 + +@ +\subsection{8.87 ${\rm cosh}(x+2k\pi i)$} +\[{\rm cosh}(x+2k\pi i) == {\rm cosh}(x)\] +<<*>>= +)clear all + +--S 236 of 298 +coshperiod:=rule(cosh(x+2*k*%pi*%i) == cosh(x)) +--R +--R (1) cosh(x + 2%i k %pi) == cosh(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 236 + +--S 237 of 298 +t1:=cosh(x+2*k*%pi*%i) - cosh(x) +--R +--R (2) cosh(x + 2%i k %pi) - cosh(x) +--R Type: Expression Complex Integer +--E 237 + +--S 238 of 298 +t2:=coshperiod t1 +--R +--R (3) cosh(x + 2%i k %pi) - cosh(x) +--R Type: Expression Complex Integer +--E 238 + +@ +\subsection{8.88 ${\rm tanh}(x+2k\pi i)$} +\[{\rm tanh}(x+k\pi i) == {\rm tanh}(x)\] +<<*>>= +)clear all + +--S 239 of 298 +tanhperiod:=rule(tanh(x+k*%pi*%i) == tanh(x)) +--R +--R (1) tanh(x + %i k %pi) == tanh(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 239 + +--S 240 of 298 +t1:=tanh(x+k*%pi*%i) - tanh(x) +--R +--R (2) tanh(x + %i k %pi) - tanh(x) +--R Type: Expression Complex Integer +--E 240 + +--S 241 of 298 +t2:=tanhperiod t1 +--R +--R (3) tanh(x + %i k %pi) - tanh(x) +--R Type: Expression Complex Integer +--E 241 + +@ +\subsection{8.89 ${\rm csch}(x+2k\pi i)$} +\[{\rm csch}(x+2k\pi i) == {\rm csch}(x)\] +<<*>>= +)clear all + +--S 242 of 298 +cschperiod:=rule(csch(x+2*k*%pi*%i) == csch(x)) +--R +--R (1) csch(x + 2%i k %pi) == csch(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 242 + +--S 243 of 298 +t1:=csch(x+2*k*%pi*%i) - csch(x) +--R +--R (2) csch(x + 2%i k %pi) - csch(x) +--R Type: Expression Complex Integer +--E 243 + +--S 244 of 298 +t2:=cschperiod t1 +--R +--R (3) csch(x + 2%i k %pi) - csch(x) +--R Type: Expression Complex Integer +--E 244 + +@ +\subsection{8.90 ${\rm sech}(x+2k\pi i)$} +\[{\rm sech}(x+2k\pi i) == {\rm sech}(x)\] +<<*>>= +)clear all + +--S 245 of 298 +sechperiod:=rule(sech(x+2*k*%pi*%i) == sech(x)) +--R +--R (1) sech(x + 2%i k %pi) == sech(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 245 + +--S 246 of 298 +t1:=sech(x+2*k*%pi*%i) - sech(x) +--R +--R (2) sech(x + 2%i k %pi) - sech(x) +--R Type: Expression Complex Integer +--E 246 + +--S 247 of 298 +t2:=sechperiod t1 +--R +--R (3) sech(x + 2%i k %pi) - sech(x) +--R Type: Expression Complex Integer +--E 247 + +@ +\subsection{8.91 ${\rm coth}(x+2k\pi i)$} +\[{\rm coth}(x+k\pi i) == {\rm coth}(x)\] +<<*>>= +)clear all + +--S 248 of 298 +cothperiod:=rule(coth(x+k*%pi*%i) == coth(x)) +--R +--R (1) coth(x + %i k %pi) == coth(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 248 + +--S 249 of 298 +t1:=coth(x+k*%pi*%i) - coth(x) +--R +--R (2) coth(x + %i k %pi) - coth(x) +--R Type: Expression Complex Integer +--E 249 + +--S 250 of 298 +t2:=cothperiod t1 +--R +--R (3) coth(x + %i k %pi) - coth(x) +--R Type: Expression Complex Integer +--E 250 + +@ + +\section{Relationship between Inverse Hyperbolic and Inverse Trigonometric Functions} +These patterns do not match in Axiom. +\subsection{8.92 $\sin^{-1}(ix)$} +\[\sin^{-1}(ix) == i~{\rm sinh}^{-1}(x)\] +<<*>>= +)clear all + +--S 251 of 298 +sinsinh:=rule(sin(%i*x)^(-1) == %i*sinh(x)^(-1)) +--R +--R 1 %i +--R (1) --------- == ------- +--R sin(%i x) sinh(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 251 + +--S 252 of 298 +t1:=sin(%i*x)^(-1) - %i*sinh(x)^(-1) +--R +--R sinh(x) - %i sin(%i x) +--R (2) ---------------------- +--R sin(%i x)sinh(x) +--R Type: Expression Complex Integer +--E 252 + +--S 253 of 298 +t2:=sinsinh t1 +--R +--R sinh(x) - %i sin(%i x) +--R (3) ---------------------- +--R sin(%i x)sinh(x) +--R Type: Expression Complex Integer +--E 253 + +@ +\subsection{8.93 ${\rm \sinh}^{-1}(ix)$} +\[{\rm \sinh}^{-1}(ix) == i~\sin^{-1}(x)\] +<<*>>= +)clear all + +--S 254 of 298 +sinhsin:=rule(sinh(%i*x)^(-1) == %i*sin(x)^(-1)) +--R +--R 1 %i +--R (1) ---------- == ------ +--R sinh(%i x) sin(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 254 + +--S 255 of 298 +t1:=sinh(%i*x)^(-1) - %i*sin(x)^(-1) +--R +--R - %i sinh(%i x) + sin(x) +--R (2) ------------------------ +--R sin(x)sinh(%i x) +--R Type: Expression Complex Integer +--E 255 + +--S 256 of 298 +t2:=sinhsin t1 +--R +--R - %i sinh(%i x) + sin(x) +--R (3) ------------------------ +--R sin(x)sinh(%i x) +--R Type: Expression Complex Integer +--E 256 + +@ +\subsection{8.94 $\cos^{-1}(x)$} +\[\cos^{-1}(x) == \pm ~{\rm cosh}^{-1}(x)\] +<<*>>= +)clear all + +--S 257 of 298 +coscosh:=rule(cos(x)^(-1) == %i*cosh(x)^(-1)) +--R +--R 1 %i +--R (1) ------ == ------- +--R cos(x) cosh(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 257 + +--S 258 of 298 +t1:=cos(x)^(-1) - %i*cosh(x)^(-1) +--R +--R cosh(x) - %i cos(x) +--R (2) ------------------- +--R cos(x)cosh(x) +--R Type: Expression Complex Integer +--E 258 + +--S 259 of 298 +t2:=coscosh t1 +--R +--R cosh(x) - %i cos(x) +--R (3) ------------------- +--R cos(x)cosh(x) +--R Type: Expression Complex Integer +--E 259 + +)clear all + +--S 260 of 298 +coscosh2:=rule(cos(x)^(-1) == -%i*cosh(x)^(-1)) +--R +--R 1 %i +--R (1) ------ == - ------- +--R cos(x) cosh(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 260 + +--S 261 of 298 +t1:=cos(x)^(-1) - -%i*cosh(x)^(-1) +--R +--R cosh(x) + %i cos(x) +--R (2) ------------------- +--R cos(x)cosh(x) +--R Type: Expression Complex Integer +--E 261 + +--S 262 of 298 +t2:=coscosh2 t1 +--R +--R cosh(x) + %i cos(x) +--R (3) ------------------- +--R cos(x)cosh(x) +--R Type: Expression Complex Integer +--E 262 + +@ +\subsection{8.95 ${\rm \cosh}^{-1}(x)$} +\[{\rm \cosh}^{-1}(x) == \pm i~\cos^{-1}(x)\] +<<*>>= +)clear all + +--S 263 of 298 +coshcos:=rule(cosh(x)^(-1) == %i*cos(x)^(-1)) +--R +--R 1 %i +--R (1) ------- == ------ +--R cosh(x) cos(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 263 + +--S 264 of 298 +t1:=cosh(x)^(-1) - %i*cos(x)^(-1) +--R +--R - %i cosh(x) + cos(x) +--R (2) --------------------- +--R cos(x)cosh(x) +--R Type: Expression Complex Integer +--E 264 + +--S 265 of 298 +t2:=coshcos t1 +--R +--R - %i cosh(x) + cos(x) +--R (3) --------------------- +--R cos(x)cosh(x) +--R Type: Expression Complex Integer +--E 265 + +)clear all + +--S 266 of 298 +coshcos2:=rule(cosh(x)^(-1) == -%i*cos(x)^(-1)) +--R +--R 1 %i +--R (1) ------- == - ------ +--R cosh(x) cos(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 266 + +--S 267 of 298 +t1:=cosh(x)^(-1) - -%i*cos(x)^(-1) +--R +--R %i cosh(x) + cos(x) +--R (2) ------------------- +--R cos(x)cosh(x) +--R Type: Expression Complex Integer +--E 267 + +--S 268 of 298 +t2:=coshcos2 t1 +--R +--R %i cosh(x) + cos(x) +--R (3) ------------------- +--R cos(x)cosh(x) +--R Type: Expression Complex Integer +--E 268 + +@ +\subsection{8.96 $\tan^{-1}(ix)$} +\[\tan^{-1}(ix) == i~{\rm tanh}^{-1}(x)\] +<<*>>= +)clear all + +--S 269 of 298 +tantanh:=rule(tan(%i*x)^(-1) == %i*tanh(x)^(-1)) +--R +--R 1 %i +--R (1) --------- == ------- +--R tan(%i x) tanh(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 269 + +--S 270 of 298 +t1:=tan(%i*x)^(-1) - %i*tanh(x)^(-1) +--R +--R tanh(x) - %i tan(%i x) +--R (2) ---------------------- +--R tan(%i x)tanh(x) +--R Type: Expression Complex Integer +--E 270 + +--S 271 of 298 +t2:=tantanh t1 +--R +--R tanh(x) - %i tan(%i x) +--R (3) ---------------------- +--R tan(%i x)tanh(x) +--R Type: Expression Complex Integer +--E 271 + +@ +\subsection{8.97 ${\rm tanh}^{-1}(ix)$} +\[{\rm tanh}^{-1}(ix) == i~\tan^{-1}(x)\] +<<*>>= +)clear all + +--S 272 of 298 +tanhtan:=rule(tanh(%i*x)^(-1) == %i*tan(x)^(-1)) +--R +--R 1 %i +--R (1) ---------- == ------ +--R tanh(%i x) tan(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 272 + +--S 273 of 298 +t1:=tanh(%i*x)^(-1) - %i*tan(x)^(-1) +--R +--R - %i tanh(%i x) + tan(x) +--R (2) ------------------------ +--R tan(x)tanh(%i x) +--R Type: Expression Complex Integer +--E 273 + +--S 274 of 298 +t2:=tanhtan t1 +--R +--R - %i tanh(%i x) + tan(x) +--R (3) ------------------------ +--R tan(x)tanh(%i x) +--R Type: Expression Complex Integer +--E 274 + +@ +\subsection{8.98 $\cot^{-1}(ix)$} +\[\cot^{-1}(ix) == -i~{\rm cosh}^{-1}(x)\] +<<*>>= +)clear all + +--S 275 of 298 +cotcoth:=rule(cot(%i*x)^(-1) == -%i*coth(x)^(-1)) +--R +--R 1 %i +--R (1) --------- == - ------- +--R cot(%i x) coth(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 275 + +--S 276 of 298 +t1:=cot(%i*x)^(-1) - -%i*coth(x)^(-1) +--R +--R coth(x) + %i cot(%i x) +--R (2) ---------------------- +--R cot(%i x)coth(x) +--R Type: Expression Complex Integer +--E 276 + +--S 277 of 298 +t2:=cotcoth t1 +--R +--R coth(x) + %i cot(%i x) +--R (3) ---------------------- +--R cot(%i x)coth(x) +--R Type: Expression Complex Integer +--E 277 + +@ +\subsection{8.99 ${\rm coth}^{-1}(ix)$} +\[{\rm coth}^{-1}(ix) == -i~\cot^{-1}(x)\] +<<*>>= +)clear all + +--S 278 of 298 +cothcot:=rule(coth(%i*x)^(-1) == -%i*cot(x)^(-1)) +--R +--R 1 %i +--R (1) ---------- == - ------ +--R coth(%i x) cot(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 278 + +--S 279 of 298 +t1:=coth(%i*x)^(-1) - -%i*cot(x)^(-1) +--R +--R %i coth(%i x) + cot(x) +--R (2) ---------------------- +--R cot(x)coth(%i x) +--R Type: Expression Complex Integer +--E 279 + +--S 280 of 298 +t2:=cothcot t1 +--R +--R %i coth(%i x) + cot(x) +--R (3) ---------------------- +--R cot(x)coth(%i x) +--R Type: Expression Complex Integer +--E 280 + +@ +\subsection{8.100 $\sec^{-1}(x)$} +\[\sec^{-1}(x) == \pm i~{\rm sech}(x)\] +<<*>>= +)clear all + +--S 281 of 298 +secsech:=rule(sec(x)^(-1) == %i*sech(x)^(-1)) +--R +--R 1 %i +--R (1) ------ == ------- +--R sec(x) sech(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 281 + +--S 282 of 298 +t1:=sec(x)^(-1) - %i*sech(x)^(-1) +--R +--R sech(x) - %i sec(x) +--R (2) ------------------- +--R sec(x)sech(x) +--R Type: Expression Complex Integer +--E 282 + +--S 283 of 298 +t2:=secsech t1 +--R +--R sech(x) - %i sec(x) +--R (3) ------------------- +--R sec(x)sech(x) +--R Type: Expression Complex Integer +--E 283 + +)clear all + +--S 284 of 298 +secsech2:=rule(sec(x)^(-1) == -%i*sech(x)^(-1)) +--R +--R 1 %i +--R (1) ------ == - ------- +--R sec(x) sech(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 284 + +--S 285 of 298 +t1:=sec(x)^(-1) - -%i*sech(x)^(-1) +--R +--R sech(x) + %i sec(x) +--R (2) ------------------- +--R sec(x)sech(x) +--R Type: Expression Complex Integer +--E 285 + +--S 286 of 298 +t2:=secsech2 t1 +--R +--R sech(x) + %i sec(x) +--R (3) ------------------- +--R sec(x)sech(x) +--R Type: Expression Complex Integer +--E 286 + +@ +\subsection{8.101 ${\rm sech}^{-1}(x)$} +\[{\rm sech}^{-1}(x) == \pm i~\sec^{-1}(x)\] +<<*>>= +)clear all + +--S 287 of 298 +sechsec:=rule(sech(x)^(-1) == %i*sec(x)^(-1)) +--R +--R 1 %i +--R (1) ------- == ------ +--R sech(x) sec(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 287 + +--S 288 of 298 +t1:=sech(x)^(-1) - %i*sec(x)^(-1) +--R +--R - %i sech(x) + sec(x) +--R (2) --------------------- +--R sec(x)sech(x) +--R Type: Expression Complex Integer +--E 288 + +--S 289 of 298 +t2:=sechsec t1 +--R +--R - %i sech(x) + sec(x) +--R (3) --------------------- +--R sec(x)sech(x) +--R Type: Expression Complex Integer +--E 289 + +)clear all + +--S 290 of 298 +sechsec:=rule(sech(x)^(-1) == -%i*sec(x)^(-1)) +--R +--R 1 %i +--R (1) ------- == - ------ +--R sech(x) sec(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 290 + +--S 291 of 298 +t1:=sech(x)^(-1) - -%i*sec(x)^(-1) +--R +--R %i sech(x) + sec(x) +--R (2) ------------------- +--R sec(x)sech(x) +--R Type: Expression Complex Integer +--E 291 + +--S 292 of 298 +t2:=sechsec t1 +--R +--R %i sech(x) + sec(x) +--R (3) ------------------- +--R sec(x)sech(x) +--R Type: Expression Complex Integer +--E 292 + +@ +\subsection{8.102 $\csc^{-1}(ix)$} +\[\csc^{-1}(ix) == -i~{\rm csch}^{-1}(x)\] +<<*>>= +)clear all + +--S 293 of 298 +csccsch:=rule(csc(%i*x)^(-1) == -%i*csch(x)^(-1)) +--R +--R 1 %i +--R (1) --------- == - ------- +--R csc(%i x) csch(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 293 + +--S 294 of 298 +t1:=csc(%i*x)^(-1) - -%i*csch(x)^(-1) +--R +--R csch(x) + %i csc(%i x) +--R (2) ---------------------- +--R csc(%i x)csch(x) +--R Type: Expression Complex Integer +--E 294 + +--S 295 of 298 +t2:=csccsch t1 +--R +--R csch(x) + %i csc(%i x) +--R (3) ---------------------- +--R csc(%i x)csch(x) +--R Type: Expression Complex Integer +--E 295 + +@ +\subsection{8.103 ${\rm csch}^{-1}(ix)$} +\[{\rm csch}^{-1}(ix) ==-i~\csc^{-1}(x)\] +<<*>>= +)clear all + +--S 296 of 298 +cschcsc:=rule(csch(%i*x)^(-1) == -%i*csc(x)^(-1)) +--R +--R 1 %i +--R (1) ---------- == - ------ +--R csch(%i x) csc(x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E 296 + +--S 297 of 298 +t1:=csch(%i*x)^(-1) - -%i*csc(x)^(-1) +--R +--R %i csch(%i x) + csc(x) +--R (2) ---------------------- +--R csc(x)csch(%i x) +--R Type: Expression Complex Integer +--E 297 + +--S 298 of 298 +t2:=cschcsc t1 +--R +--R %i csch(%i x) + csc(x) +--R (3) ---------------------- +--R csc(x)csch(%i x) +--R Type: Expression Complex Integer +--E 298 + +@ + +\begin{verbatim} +rootrule4a:RewriteRule(INT,COMPLEX(INT),EXPR(COMPLEX(INT))):=rule(sqrt(p^2-q^2)==sqrt(p-q)*sqrt(q+p)) schaum17.input.pamphlet: +tanrule2:RewriteRule(INT,COMPLEX(INT),EXPR(COMPLEX(INT))):=rule(tan(a) == sin(a)/cos(a)) schaum17.input.pamphlet: +\end{verbatim} + +<<*>>= +)spool +)lisp (bye) +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document}