# integrals of modified Bessel functions

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##### 2: 10.1 Special Notation
For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
##### 5: 6.10 Other Series Expansions
###### §6.10(ii) Expansions in Series of Spherical BesselFunctions
6.10.6 $\operatorname{Ei}\left(x\right)=\gamma+\ln\left|x\right|+\sum_{n=0}^{\infty}(-% 1)^{n}(x-a_{n})\left({\mathsf{i}^{(1)}_{n}}\left(\tfrac{1}{2}x\right)\right)^{% 2},$ $x\neq 0$,
6.10.8 $\operatorname{Ein}\left(z\right)=ze^{-z/2}\left({\mathsf{i}^{(1)}_{0}}\left(% \tfrac{1}{2}z\right)+\sum_{n=1}^{\infty}\dfrac{2n+1}{n(n+1)}{\mathsf{i}^{(1)}_% {n}}\left(\tfrac{1}{2}z\right)\right).$
##### 9: 8.6 Integral Representations
8.6.6 $\Gamma\left(a,z\right)=\frac{2z^{\frac{1}{2}a}e^{-z}}{\Gamma\left(1-a\right)}% \int_{0}^{\infty}e^{-t}t^{-\frac{1}{2}a}K_{a}\left(2\sqrt{zt}\right)\,\mathrm{% d}t,$ $\Re a<1$,
##### 10: Bibliography T
• N. M. Temme (1990b) Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions. SIAM J. Math. Anal. 21 (1), pp. 241–261.
• N. M. Temme (1994c) Steepest descent paths for integrals defining the modified Bessel functions of imaginary order. Methods Appl. Anal. 1 (1), pp. 14–24.
• N. M. Temme (1978) The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions. Report TW 183/78 Mathematisch Centrum, Amsterdam, Afdeling Toegepaste Wiskunde.