As a general reference to the functions described in this section (and much more), we refer the reader to Whittaker and Watson [WW15].
Return the Gamma function Gamma(f) of the series f. f must be defined over the free real or complex field, the valuation of f must be 0 and the constant term of f must be 1.
Given a positive real number of fixed precision, or a free real or complex number s (not equal to 0, - 1, - 2, ... ), calculate the value Gamma(s) of the gamma function at s. For s with positive real part this is the value of the integral from 0 to infinity of u^(x - 1)e^(-u) with respect to u. For other s (not a non-positive integer) the function is defined by analytic continuation, and it satisfies the product formula (1/s Gamma(s))=e^(gamma s)prod_(n=1)^Infinity (1 + (s/n))e^(-s/n). The function Gamma also satisfies Gamma(s)Gamma(1 - s)=(pi/(sin(pi s))), and Gamma(s + 1)=s Gamma(s).
Complementary: BoolElt Default: false
Gamma: FldPrElt Default:
For free or fixed precision real numbers s, t this returns the value of the incomplete gamma function gamma(s, t)=int_0^t u^(s - 1)e^(-u)du. The optional argument Complementary can be used to find the complement int_t^Infinity u^(s - 1)e^(-u)du instead. If the arguments are free real numbers, there is a second optional argument that may be used in the computation of the incomplete gamma value; the free real value of Gamma should be the value of Gamma(s), in which case gamma(s, t) may be computed as the difference between the given value for Gamma(s) and that of the complementary gamma at s, t.
For free real s (such that s + (1/2) is not a non-positive integer) this returns the value of Gamma(s + (1/2)). For integer values of s this is faster than Gamma(s+(1/2)), because Legendre's doubling formula Gamma(s + (1/2))=2^(1 - 2s)Sqrt(pi)(Gamma(2s)/Gamma(s)) is used.
Return the Log-Gamma function ( Log)(Gamma(f)) of the series f. f must be defined over the free real or complex field, the valuation of f must be 0 and the constant term of f must be 1.
For fixed precision positive real s or free real or complex s (not a non-positive integer) return the value of the principal branch of the logarithm of the gamma function of s.
For free real or complex s (not a non-positive integer) return the principal value of the logarithmic derivative Psi(s)=(d log Gamma(s)/ds)=(Gamma'(s)/Gamma(s)), of the gamma function, which allows the expansion Psi(s)= - gamma - (1/s) + s sum_(n=1)^Infinity(1/n(s + n)); here gamma is Euler's gamma.
Given a small integer n and a real number r of fixed precision, calculate the value of the Bessel function y = J_n(r), of the first kind of order n. Results for negative arguments are defined by: J_(-n)(r) = J_n( - r) = ( - 1)^n J_n(r). The Bessel function of the first kind of order n is defined by J_n(x)=(1/2pi i)((z/2))^n int_(-Infinity)^(0^ + ) u^(-n - 1)e^(u - (z^2/4t))du, and satisfies J_n(x)=sum_(k=0)^Infinity (( - 1)^kz^(n + 2k)/2^(n + 2k)k!Gamma(n + k + 1)). The method used is Hankel's asymptotic expansion if abs(r) is large, the power series if abs(r) is small, and the backward recurrence method otherwise.
Given a small integer n and a free real or complex number s, calculate the value of the Bessel function of the first kind of half integral index n + (1/2), J_(n + (1/2)), defined as above.
Given a free complex n and a free positive real s, compute the value of the modified Bessel function of the second kind K_n(s), which may be defined by K_n(s)=(pi/2)(i^nJ_(-n)(i s) - i^(-n)J_n(s))cot(npi). The function KBessel2 is an alternative (often faster) implementation of this function.[Next][Prev] [Right] [Left] [Up] [Index] [Root]