### Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials

Richard Barakat
1961 Mathematics of Computation
During the course of some work on the diffraction theory of aberrations it was necessary to evaluate numerically the incomplete gamma function of imaginary argument y(v, ix) for certain values of the parameter v. These integrals are special cases of the confluent hypergeometric function, and in the standard notation  .ix y(v, ix) = / e~'t' ' dt 7?e(") > 0 (1) Jo where XFX is the confluent hypergeometric function. Only the case of v real is considered. When v is an integer, iFi is simply a
more » ... iFi is simply a polynomial in x. It is also possible to evaluate (1) in terms of the Fresnel integrals when (v -1) is a half-integer. For v = \, (1) is proportional to the Fresnel integral. For other half-integer values, we can use the recurrence relation (2) 7(1 + ", ix) = vy(", ix) -(ix)'e~ix to generate the necessary formulas. An alternate procedure utilizes the Lommel functions of two variables  ; unfortunately, the Lommel functions have not been extensively tabulated. The remaining values of v can be treated by a Taylor series expansion for small values of x and by asymptotic developments for large values of x. The main difficulty is the intermediate range where x is neither large nor small. A possible method is to use Nielsen's  representation (3) y{v, ix + iy) -y{v, ix) = e~ix(ix)'-1 ¿ (1 -,)"( -w)-"[l -e'^dy)] 71=0 where en(iy) is the truncated exponential series (4) en(iy) = ± (iyY" ¿Zo ml This method seems unnecessarily complicated. A more powerful method is to utilize the Chebyshev polynomials. Instead of expanding the exponential in ( 1 ) into a Taylor series, we expand into a series of Chebyshev polynomials. As Lanczos  has pointed out, "While the expansion into powers on the basis of the Taylor's series gives the slowest convergence, the expansion into the Chebyshev polynomials gives the fastest convergence." It is convenient to transform the integral slightly by setting t -ixq in- ( 1 )