diff --git a/changelog b/changelog index d14449f..7966a8b 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,4 @@ +20080302 tpd src/hyper/bookvol11 add additional hyperdoc page translations 20080301 tpd src/hyper/bookvol11 add additional hyperdoc page translations 20080229 tpd src/hyper/bookvol11 add additional hyperdoc page translations 20080222 tpd src/Makefile move hyperdoc bitmaps location diff --git a/src/hyper/bookvol11.pamphlet b/src/hyper/bookvol11.pamphlet index f3d863e..4871419 100644 --- a/src/hyper/bookvol11.pamphlet +++ b/src/hyper/bookvol11.pamphlet @@ -473,6 +473,7 @@ PAGES=rootpage.xhtml \ dbopasech.xhtml \ dbopatan.xhtml \ dbopatanh.xhtml \ + dbopbernoullib.xhtml \ dbopbesseli.xhtml \ dbopbesselj.xhtml \ dbopbesselk.xhtml \ @@ -480,6 +481,8 @@ PAGES=rootpage.xhtml \ dbopbeta.xhtml \ dbopbinary.xhtml \ dbopcardinalnumber.xhtml \ + dbopchebyshevt.xhtml \ + dbopchebyshevu.xhtml \ dbopcoefficient.xhtml \ dbopcoefficients.xhtml \ dbopcoerce.xhtml \ @@ -503,11 +506,13 @@ PAGES=rootpage.xhtml \ dbopcsc.xhtml \ dbopcsch.xhtml \ dbopcycleragits.xhtml \ + dbopcyclotomic.xhtml \ dbopd.xhtml \ dbopdecimal.xhtml \ dbopdefiningpolynomial.xhtml \ dbopdegree.xhtml \ dbopdenom.xhtml \ + dbopdraw.xhtml \ dbopdeterminant.xhtml \ dbopdiagonalmatrix.xhtml \ dbopdigamma.xhtml \ @@ -522,6 +527,7 @@ PAGES=rootpage.xhtml \ dbopeigenvectors.xhtml \ dbopelt.xhtml \ dbopequal.xhtml \ + dbopeulere.xhtml \ dbopeulerphi.xhtml \ dbopeval.xhtml \ dbopevenq.xhtml \ @@ -537,6 +543,7 @@ PAGES=rootpage.xhtml \ dbopfractionpart.xhtml \ dbopgamma.xhtml \ dbopgcd.xhtml \ + dbophermiteh.xhtml \ dbophex.xhtml \ dbophorizconcat.xhtml \ dbophtrigs.xhtml \ @@ -546,6 +553,7 @@ PAGES=rootpage.xhtml \ dbopinverse.xhtml \ dbopinvmod.xhtml \ dbopjacobi.xhtml \ + dboplaguerrel.xhtml \ dboplaurent.xhtml \ dboplcm.xhtml \ dbopleadingcoefficient.xhtml \ @@ -8224,6 +8232,16 @@ the operations will have extra ones added at some stage. <> @ +\subsection{dbopbernoullib.xhtml} +<>= +<> + + +<> +dbopbernoullib not implemented +<> +@ + \subsection{dbopbesseli.xhtml} <>= <> @@ -8284,6 +8302,26 @@ the operations will have extra ones added at some stage. <> @ +\subsection{dbopchebyshevt.xhtml} +<>= +<> + + +<> +dbopchebyshevt not implemented +<> +@ + +\subsection{dbopchebyshevu.xhtml} +<>= +<> + + +<> +dbopchebyshevu not implemented +<> +@ + \subsection{dbopcoefficient.xhtml} <>= <> @@ -8514,6 +8552,16 @@ the operations will have extra ones added at some stage. <> @ +\subsection{dbopcyclotomic.xhtml} +<>= +<> + + +<> +dbopcyclotomic not implemented +<> +@ + \subsection{dbopd.xhtml} <>= <> @@ -8564,6 +8612,16 @@ the operations will have extra ones added at some stage. <> @ +\subsection{dbopdraw.xhtml} +<>= +<> + + +<> +dbopdraw not implemented +<> +@ + \subsection{dbopdeterminant.xhtml} <>= <> @@ -8704,6 +8762,16 @@ the operations will have extra ones added at some stage. <> @ +\subsection{dbopeulere.xhtml} +<>= +<> + + +<> +dbopeulere not implemented +<> +@ + \subsection{dbopeulerphi.xhtml} <>= <> @@ -8854,6 +8922,16 @@ dbopfractionpart not implemented <> @ +\subsection{dbophermiteh.xhtml} +<>= +<> + + +<> +dbophermiteh not implemented +<> +@ + \subsection{dbophex.xhtml} <>= <> @@ -8945,6 +9023,16 @@ dbopfractionpart not implemented <> @ +\subsection{dboplaguerrel.xhtml} +<>= +<> + + +<> +dboplaguerrel not implemented +<> +@ + \subsection{dboplaurent.xhtml} <>= <> @@ -38490,6 +38578,10 @@ Although they have somewhat limited utility, Axiom provides Roman numerals. \subsection{numnumericfunctions.xhtml} <>= <> + <> @@ -38687,6 +38779,318 @@ of the functions yield an error if the result is not real.
+A number of additional operations may be used to compute numerical +values. These are special polynomial functions that can be evaluated +for values in any commutative ring R, and in particular for values in +any floating-point type. The following operations are provided by the +package +OrthogonalPolynomialFunctions: +
    +
  • chebyshevT: + (nonNegativeInteger,R) -> R +
    + chebyshevT(n,z) is the nth Chebyshev polynomial of the first kind, + T[n](z). These are defined by +
    + (1-t*z)/(1-2*t*z*t**2)=sum(T[n](z)*t**n,n=0..) +
    • +
  • chebyshevU: + (nonNegativeInteger,R) -> R +
    + chebyshevU(n,z) is the nth Chebyshev polynomial of the second kind, + U[n](z). These are defined by +
    + 1/(1-2*t*z+t**2)=sum(U[n](z)*t**n,n=0..) +
    • +
  • hermiteH: + (NonNegativeInteger,R) -> R +
    + hermiteH(n,z) is the nth Hermite polynomial, H[n](z). These are + defined by +
    + exp(2*t*z-t**2)=sum(H[n](z)*t**n/n!,n=0..) +
    • +
  • laguerreL: + (NonNegativeInteger,R) -> R +
    + laguerreL(n,z) is the nth Laguerre polynomial, L[n](z). These are + defined by +
    + (exp(-t*z/(1-t))/(1-t)=sum(L[n](z)*t**n/n!,n=0..) +
    • +
  • laguerreL: + (NonNegativeInteger,NonNegativeInteger,R) -> R +
    + labuerreL(m,n,2) is the associated Laguerre polynomial, L<m>[n](z). + This is the nth derivative of L[n](z). +
    • +
  • legendreP: + (NonNegativeInteger,R) -> R +
    + legendreP(n,z) is the nth Legendre polynomial, P[n](z). These are + defined by +
    + 1/sqrt(1-2*z*t+t**2)=sum(P[n](z)*t**n,n=0..) +
    • +
+
+
+These operations require non-negative integers for the indices, +but otherwise the argument can be given as desired. +
    +
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    • +
+The expression chebyshevT(n,z) evaluates to the nth Chebyshev polynomial +of the first kind. +
    +
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+The expression chebyshevU(n,z) evaluates to the nth Chebyshev polynomial +of the second kind. +
    +
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    • +
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+The expression hermiteH(n,z) evaluates to the nth Hermite polynomial. +
    +
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    • +
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+The expression laguerreL(n,z) evaluates to the nth Laguerre polynomial. +
    +
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+The expression legendreP(n,z) evaluates to the nth Legendre polynomial. +
    +
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    • +
+
+
+Finally, three number-theoretic polynomial operations may be evaluated. +The following operations are provided by the package + +NumberTheoreticPolynomialFunctions. +
    +
  • bernoulliB: + (NonNegativeInteger,R) -> R +
    + bernoulliB(n,z) is the nth Bernoulli polynomial, B[n](z). These are + defined by +
    + t*exp(z*t)/(exp t - 1)=sum(B[n](z)*t**n/n! for n=0..) +
    • +
  • eulerE: + (NonNegativeInteger,R) -> R +
    + eulerE(n,z) is the nth Euler polynomial, E[n](z). These are defined by +
    + 2*exp(z*t)/(exp t + 1)=sum(E[n](z)*t**n/n! for n=0..) +
    • +
  • cyclotomic: + (NonNegativeInteger,R) -> R +
    + cyclotomic(n,z) is the nth cyclotomic polynomial φ(n,z). + This is the polynomial whose roots are precisely the primitive nth + roots of unity. This polynomial has degree given by the Euler + totient function φ(n). +
    • +
+ +The expression bernoulliB(n,z) evaluates to the nth Bernoulli polynomial. +
    +
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+The expression eulerE(n,z) evaluates to the nth Euler polynomial. +
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+The expression cyclotomic(n,z) evaluates to the nth cyclotomic polynomial. +
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+
+
+Drawing complex functions in Axiom is presently somewhat awkward compared +to drawing real functions. It is necessary to use the +draw operations that operate on functions +rather than expressions. + +This is the complex exponential function. When this is displayed in color, +the height is the value of the real part of the function and the color is +the imaginary part. Red indicates large negative imaginary values, green +indicates imaginary values near zero and blue/violet indicates large +positive imaginary values. +
    +
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+This is the complex arctangent function. Again, the height is the real part +of the function value but here the color indicates the function value's phase. +The position of the branch cuts are clearly visible and one can see that the +function is real only for a real argument. +
    +
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+This is the complex Gamma function. +
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+This shows the real Beta function near the origin. +
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+This is the Bessel function J(alpha,x) for index alpha in the range -6..4 and +argument x in the range 2..14. +
    +
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+This is the modified Bessel function I(alpha,x) evaluated for various real +values of the index alpha and fixed argument x=5. +
    +
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+This is similar to the last example except the index alpha takes on complex +values in a 6x6 rectangle centered on the origin. +
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<> @