Extra C99 double complex transcendental functions with Arb
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Extra C99 double complex transcendental functions with Arb.

Author: Fredrik Johansson (fredrik.johansson@gmail.com)

This code is public domain.


C99 provides support for complex numbers via the standard library header complex.h, but only includes a handful of transcendental functions.

This is a simple wrapper of Arb (https://github.com/fredrik-johansson/arb/), exposing many more useful complex transcendental functions (Riemann zeta, polylogarithm, Bessel, incomplete gamma, hypergeometric, Jacobi theta, etc.) in double precision.

All function arguments as well as the return value have the C99 double complex type. The output, if finite, is guaranteed to have a relative error that is at most a small multiple of 2^-53 (todo: specify an explicit multiple here), unless the function value is so small that this value underflows to zero. An infinite/NaN return value indicates either that the evaluation has failed to convergence (possibly due to trying to evaluate the function at a point where it is undefined).

Currently, Arb 2.8.0 (or the git master of Arb) is required.

Provided functions

ac_exp(z)               Exponential function
ac_expm1(z)             Accurate exp(z)-1
ac_log(z)               Natural logarithm
ac_log1p(z)             Accurate log(1+z)
ac_sqrt(z)              Square root
ac_rsqrt(z)             Reciprocal square root
ac_cbrt(z)              Cube root
ac_pow(a,b)             Power a^b

ac_sin(z)               Trigonometric functions
ac_sinpi(z)             Trigonometric functions, argument multiplied by pi
ac_asin(z)              Inverse trigonometric functions
ac_sinh(z)              Hyperbolic functions
ac_asinh(z)             Inverse hyperbolic functions

ac_gamma(z)             Gamma function
ac_rgamma(z)            Reciprocal gamma function
ac_lgamma(z)            Logarithmic gamma function
ac_digamma(z)           Digamma function
ac_zeta(s)              Riemann zeta function
ac_zeta2(s,a)           Hurwitz zeta function
ac_polygamma(s,z)       Polygamma function
ac_polylog(s,z)         Polylogarithm
ac_barnesg(s)           Barnes G-function
ac_lbarnesg(s)          Logarithmic Barnes G-function

ac_erf(s)               Error function
ac_erfc(s)              Complementary error function
ac_erfi(s)              Imaginary error function
ac_gammaup(s,z)         Upper incomplete gamma function
ac_expint(s,z)          Generalized exponential integral E
ac_ei(z)                Exponential integral Ei
ac_si(z)                Sine integral
ac_ci(z)                Cosine integral
ac_shi(z)               Hyperbolic sine integral
ac_chi(z)               Hyperbolic cosine integral
ac_li(z)                Logarithmic integral
ac_lioffset(z)          Offset logarithmic integral

ac_besselj(v,z)         Bessel function J
ac_bessely(v,z)         Bessel function Y
ac_besseli(v,z)         Bessel function I
ac_besselk(v,z)         Bessel function K
ac_ai(z)                Airy function Ai
ac_aiprime(z)           Airy function derivative Ai'
ac_bi(z)                Airy function Bi
ac_biprime(z)           Airy function derivative Bi'

ac_hyperu(a,b,z)        Confluent hypergeometric function U
ac_hyp0f1(a,z)          Confluent hypergeometric function 0F1
ac_hyp0f1r(a,z)         Regularized confluent hypergeometric function 0F1
ac_hyp1f1(a,b,z)        Confluent hypergeometric function 1F1
ac_hyp1f1r(a,b,z)       Regularized confluent hypergeometric function 1F1
ac_hyp2f1(a,b,c,z)      Hypergeometric function 2F1
ac_hyp2f1r(a,b,c,z)     Regularized hypergeometric function 2F1

ac_chebyt(n,z)          Chebyshev polynomial/function T
ac_chebyu(n,z)          Chebyshev polynomial/function U
ac_jacobip(n,a,b,z)     Jacobi polynomial/function P
ac_gegenbauerc(n,m,z)   Gegenbauer polynomial/function C
ac_laguerrel(n,m,z)     Laguerre polynomial/function L
ac_hermiteh(n,z)        Hermite polynomial/function H
ac_legenp(n,m,z)        Associated Legendre polynomial/function P
ac_legenpv(n,m,z)       Associated Legendre polynomial/function P (alt. branch)
ac_legenq(n,m,z)        Associated Legendre polynomial/function Q
ac_legenqv(n,m,z)       Associated Legendre polynomial/function Q (alt. branch)

ac_modeta(tau)          Dedekind eta function
ac_modj(tau)            Modular j-invariant
ac_modlambda(tau)       Modular lambda function
ac_moddelta(tau)        Modular delta function
ac_agm1(z)              Arithmetic-geometric mean of 1 and z
ac_ellipk(z)            Complete elliptic integral K
ac_ellipe(z)            Complete elliptic integral E
ac_ellipp(z,tau)        Weierstrass elliptic function P
ac_theta1(z,tau)        Jacobi theta function theta1
ac_theta2(z,tau)        Jacobi theta function theta2
ac_theta3(z,tau)        Jacobi theta function theta3
ac_theta4(z,tau)        Jacobi theta function theta4

Development ideas

Make it a library (don't define everything as inline functions in a header file).

Write some test code.

Write a similar module for real-valued functions.

Write a similar module for long double, GCC quadruple precision, etc.

Allow the user to set different output tolerances. For example, one could guarantee correct rounding (requires knowing the exact input-output pairs of the function), or guarantee that both the real and imaginary parts are accurate separately (requires knowing where the function is exactly real/imaginary). The user might also want to lower the precision (say, guaranteeing only a relative error of 1e-8) as a tradeoff for speed.

Allow more fine-grained control over what to do when convergence fails, when the final conversion overflows/underflows the exponent range of a double, etc.

Wrap the Arb functions that compute several functions or function derivatives simultaneously, e.g. sin+cos, Bessel J+Y, Jacobi theta 1+2+3+4.

Wrap Arb functions that take integer parameters as inputs (e.g. nth root).