# along the real line

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## 1—10 of 18 matching pages

##### 3: 10.32 Integral Representations
###### §10.32(i) Integrals along the RealLine
10.32.11 $K_{\nu}\left(xz\right)=\frac{\Gamma\left(\nu+\frac{1}{2}\right)(2z)^{\nu}}{\pi% ^{\frac{1}{2}}x^{\nu}}\int_{0}^{\infty}\frac{\cos\left(xt\right)\,\mathrm{d}t}% {(t^{2}+z^{2})^{\nu+\frac{1}{2}}},$ $\Re\nu>-\tfrac{1}{2}$, $x>0$, $|\operatorname{ph}z|<\tfrac{1}{2}\pi$.
##### 7: 10.9 Integral Representations
###### §10.9(i) Integrals along the RealLine
10.9.11 ${H^{(2)}_{\nu}}\left(z\right)=-\frac{e^{\frac{1}{2}\nu\pi i}}{\pi i}\int_{-% \infty}^{\infty}e^{-iz\cosh t-\nu t}\,\mathrm{d}t,$ $-\pi<\operatorname{ph}z<0$.
10.9.16 $\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}{H^{(2)}_{\nu}}\left((z^{% 2}-\zeta^{2})^{\frac{1}{2}}\right)=-\frac{1}{\pi i}e^{\frac{1}{2}\nu\pi i}\int% _{-\infty}^{\infty}e^{-iz\cosh t-i\zeta\sinh t-\nu t}\,\mathrm{d}t,$ $\Im\left(z\pm\zeta\right)<0$.
##### 8: 25.5 Integral Representations
###### §25.5 Integral Representations
25.5.19 $\zeta\left(m+s\right)=(-1)^{m-1}\frac{\Gamma\left(s\right)\sin\left(\pi s% \right)}{\pi\Gamma\left(m+s\right)}\*\int_{0}^{\infty}{\psi}^{(m)}\left(1+x% \right)x^{-s}\,\mathrm{d}x,$ $m=1,2,3,\dots$.
##### 10: 4.3 Graphics
###### §4.3(i) Real Arguments
Figure 4.3.2 illustrates the conformal mapping of the strip $-\pi<\Im z<\pi$ onto the whole $w$-plane cut along the negative real axis, where $w=e^{z}$ and $z=\ln w$ (principal value). …Lines parallel to the real axis in the $z$-plane map onto rays in the $w$-plane, and lines parallel to the imaginary axis in the $z$-plane map onto circles centered at the origin in the $w$-plane. In the labeling of corresponding points $r$ is a real parameter that can lie anywhere in the interval $(0,\infty)$. …