# auxiliary functions

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## 11—20 of 30 matching pages

##### 11: 7.7 Integral Representations
###### §7.7(ii) AuxiliaryFunctions
7.7.12 $\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=e^{-\pi iz^{2}/2}\int_{z}^{% \infty}e^{\pi it^{2}/2}\mathrm{d}t.$
##### 13: 6.2 Definitions and Interrelations
###### §6.2(iii) AuxiliaryFunctions
6.2.17 $\mathrm{f}\left(z\right)=\phantom{+}\mathrm{Ci}\left(z\right)\sin z-\mathrm{si% }\left(z\right)\cos z,$
6.2.18 $\mathrm{g}\left(z\right)=-\mathrm{Ci}\left(z\right)\cos z-\mathrm{si}\left(z% \right)\sin z.$
6.2.19 $\mathrm{Si}\left(z\right)=\tfrac{1}{2}\pi-\mathrm{f}\left(z\right)\cos z-% \mathrm{g}\left(z\right)\sin z,$
6.2.20 $\mathrm{Ci}\left(z\right)=\mathrm{f}\left(z\right)\sin z-\mathrm{g}\left(z% \right)\cos z.$
##### 14: 6.12 Asymptotic Expansions
6.12.5 $\mathrm{f}\left(z\right)=\frac{1}{z}\sum_{m=0}^{n-1}(-1)^{m}\frac{(2m)!}{z^{2m% }}+R_{n}^{(\mathrm{f})}(z),$
##### 15: 8.21 Generalized Sine and Cosine Integrals
###### §8.21(vii) AuxiliaryFunctions
8.21.18 $f(a,z)=\mathrm{si}\left(a,z\right)\cos z-\mathrm{ci}\left(a,z\right)\sin z,$
8.21.20 $\mathrm{si}\left(a,z\right)=f(a,z)\cos z+g(a,z)\sin z,$
8.21.21 $\mathrm{ci}\left(a,z\right)=-f(a,z)\sin z+g(a,z)\cos z.$
##### 16: 6.7 Integral Representations
###### §6.7(iii) AuxiliaryFunctions
6.7.12 $\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=e^{-iz}\int_{z}^{\infty}% \frac{e^{it}}{t}\mathrm{d}t,$ $|\operatorname{ph}z|\leq\pi.$
6.7.13 $\mathrm{f}\left(z\right)=\int_{0}^{\infty}\frac{\sin t}{t+z}\mathrm{d}t=\int_{% 0}^{\infty}\frac{e^{-zt}}{t^{2}+1}\mathrm{d}t,$
6.7.15 $\mathrm{f}\left(z\right)=2\int_{0}^{\infty}K_{0}\left(2\sqrt{zt}\right)\cos t% \mathrm{d}t,$
6.7.16 $\mathrm{g}\left(z\right)=2\int_{0}^{\infty}K_{0}\left(2\sqrt{zt}\right)\sin t% \mathrm{d}t.$
##### 17: 8.24 Physical Applications
With more general values of $p$, $E_{p}\left(x\right)$ supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).
##### 18: 12.19 Tables
• Miller (1955) includes $W\left(a,x\right)$, $W\left(a,-x\right)$, and reduced derivatives for $a=-10(1)10$, $x=0(.1)10$, 8D or 8S. Modulus and phase functions, and also other auxiliary functions are tabulated.

• Fox (1960) includes modulus and phase functions for $W\left(a,x\right)$ and $W\left(a,-x\right)$, and several auxiliary functions for $x^{-1}=0(.005)0.1$, $a=-10(1)10$, 8S.

##### 20: 7.12 Asymptotic Expansions
###### §7.12(ii) Fresnel Integrals
7.12.2 $\mathrm{f}\left(z\right)\sim\frac{1}{\pi z}\sum_{m=0}^{\infty}(-1)^{m}\frac{{% \left(\tfrac{1}{2}\right)_{2m}}}{(\pi z^{2}/2)^{2m}},$
7.12.3 $\mathrm{g}\left(z\right)\sim\frac{1}{\pi z}\sum_{m=0}^{\infty}(-1)^{m}\frac{{% \left(\tfrac{1}{2}\right)_{2m+1}}}{(\pi z^{2}/2)^{2m+1}},$
7.12.4 $\mathrm{f}\left(z\right)=\frac{1}{\pi z}\sum_{m=0}^{n-1}(-1)^{m}\frac{{\left(% \tfrac{1}{2}\right)_{2m}}}{(\pi z^{2}/2)^{2m}}+R_{n}^{(\mathrm{f})}(z),$
7.12.5 $\mathrm{g}\left(z\right)=\frac{1}{\pi z}\sum_{m=0}^{n-1}(-1)^{m}\frac{{\left(% \tfrac{1}{2}\right)_{2m+1}}}{(\pi z^{2}/2)^{2m+1}},+R_{n}^{(\mathrm{g})}(z),$