# reciprocal-modulus transformation

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## 11—20 of 161 matching pages

##### 12: 16.20 Integrals and Series
Extensive lists of Laplace transforms and inverse Laplace transforms of the Meijer $G$-function are given in Prudnikov et al. (1992a, §3.40) and Prudnikov et al. (1992b, §3.38). …
##### 13: 16.6 Transformations of Variable
###### Cubic
16.6.2 ${{}_{3}F_{2}}\left({a,2b-a-1,2-2b+a\atop b,a-b+\frac{3}{2}};\frac{z}{4}\right)% =(1-z)^{-a}{{}_{3}F_{2}}\left({\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{% 3}a+\frac{2}{3}\atop b,a-b+\frac{3}{2}};\frac{-27z}{4(1-z)^{3}}\right).$
For Kummer-type transformations of ${{}_{2}F_{2}}$ functions see Miller (2003) and Paris (2005a), and for further transformations see Erdélyi et al. (1953a, §4.5), Miller and Paris (2011), Choi and Rathie (2013) and Wang and Rathie (2013).
##### 14: 17.18 Methods of Computation
###### §17.18 Methods of Computation
The two main methods for computing basic hypergeometric functions are: (1) numerical summation of the defining series given in §§17.4(i) and 17.4(ii); (2) modular transformations. …
##### 15: 27.17 Other Applications
###### §27.17 Other Applications
Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. …
##### 16: 19.13 Integrals of Elliptic Integrals
###### §19.13(iii) Laplace Transforms
For direct and inverse Laplace transforms for the complete elliptic integrals $K\left(k\right)$, $E\left(k\right)$, and $D\left(k\right)$ see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.
##### 17: 3.9 Acceleration of Convergence
###### §3.9(ii) Euler’s Transformation of Series
Euler’s transformation is usually applied to alternating series. …
##### 18: 32.7 Bäcklund Transformations
###### §32.7 Bäcklund Transformations
Then the transformationsThe quadratic transformation …The quartic transformation
##### 19: 2.6 Distributional Methods
###### §2.6(ii) Stieltjes Transform
The Stieltjes transform of $f(t)$ is defined by … $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ being the Mellin transform of $f(t)$ or its analytic continuation (§2.5(ii)). … Corresponding results for the generalized Stieltjes transformwhere $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ is the Mellin transform of $f$ or its analytic continuation. …
##### 20: 7.14 Integrals
###### Laplace Transforms
7.14.2 $\int_{0}^{\infty}e^{-at}\operatorname{erf}\left(bt\right)\mathrm{d}t=\frac{1}{% a}e^{a^{2}/(4b^{2})}\operatorname{erfc}\left(\frac{a}{2b}\right),$ $\Re a>0$, $|\operatorname{ph}b|<\tfrac{1}{4}\pi$,
7.14.3 $\int_{0}^{\infty}e^{-at}\operatorname{erf}\sqrt{bt}\mathrm{d}t=\frac{1}{a}% \sqrt{\frac{b}{a+b}},$ $\Re a>0$, $\Re b>0$,