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21—30 of 410 matching pages

21: 4.8 Identities
4.8.2 $\ln\left(z_{1}z_{2}\right)=\ln z_{1}+\ln z_{2},$ $-\pi\leq\operatorname{ph}z_{1}+\operatorname{ph}z_{2}\leq\pi$,
4.8.12 $\ln\left(a^{z}\right)=z\ln a+2k\pi\mathrm{i},$
4.8.15 $a^{z}b^{z}=(ab)^{z},$ $-\pi\leq\operatorname{ph}a+\operatorname{ph}b\leq\pi$,
22: 7.2 Definitions
7.2.1 $\operatorname{erf}z=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}\mathrm{d}t,$
7.2.7 $C\left(z\right)=\int_{0}^{z}\cos\left(\tfrac{1}{2}\pi t^{2}\right)\mathrm{d}t,$
7.2.8 $S\left(z\right)=\int_{0}^{z}\sin\left(\tfrac{1}{2}\pi t^{2}\right)\mathrm{d}t,$
23: 7.6 Series Expansions
7.6.10 $C\left(z\right)=z\sum_{n=0}^{\infty}\mathsf{j}_{2n}\left(\tfrac{1}{2}\pi z^{2}% \right),$
24: 25.4 Reflection Formulas
25.4.4 $\xi\left(s\right)=\tfrac{1}{2}s(s-1)\Gamma\left(\tfrac{1}{2}s\right)\pi^{-s/2}% \zeta\left(s\right).$
25.4.5 $(-1)^{k}{\zeta}^{(k)}\left(1-s\right)=\frac{2}{(2\pi)^{s}}\sum_{m=0}^{k}\sum_{% r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\left(\Re% \left(c^{k-m}\right)\cos\left(\tfrac{1}{2}\pi s\right)+\Im\left(c^{k-m}\right)% \sin\left(\tfrac{1}{2}\pi s\right)\right){\Gamma}^{(r)}\left(s\right){\zeta}^{% (m-r)}\left(s\right),$
25: 7.5 Interrelations
7.5.1 $F\left(z\right)=\tfrac{1}{2}i\sqrt{\pi}\left(e^{-z^{2}}-w\left(z\right)\right)% =-\tfrac{1}{2}i\sqrt{\pi}e^{-z^{2}}\operatorname{erf}\left(iz\right).$
7.5.3 $C\left(z\right)=\tfrac{1}{2}+\mathrm{f}\left(z\right)\sin\left(\tfrac{1}{2}\pi z% ^{2}\right)-\mathrm{g}\left(z\right)\cos\left(\tfrac{1}{2}\pi z^{2}\right),$
7.5.5 $e^{-\frac{1}{2}\pi iz^{2}}\mathcal{F}\left(z\right)=\mathrm{g}\left(z\right)+i% \mathrm{f}\left(z\right).$
7.5.7 $\zeta=\tfrac{1}{2}\sqrt{\pi}(1\mp i)z,$
7.5.13 $G\left(x\right)=\sqrt{\pi}F\left(x\right)-\tfrac{1}{2}e^{-x^{2}}\mathrm{Ei}% \left(x^{2}\right),$ $x>0$.
26: 11.4 Basic Properties
11.4.6 $\mathbf{H}_{-\frac{1}{2}}\left(z\right)=\left(\frac{2}{\pi z}\right)^{\frac{1}% {2}}\sin z,$
11.4.7 $\mathbf{L}_{\frac{1}{2}}\left(z\right)=\left(\frac{2}{\pi z}\right)^{\frac{1}{% 2}}(\cosh z-1),$
11.4.8 $\mathbf{L}_{-\frac{1}{2}}\left(z\right)=\left(\frac{2}{\pi z}\right)^{\frac{1}% {2}}\sinh z,$
11.4.10 $\mathbf{H}_{-\frac{3}{2}}\left(z\right)=\left(\frac{2}{\pi z}\right)^{\frac{1}% {2}}\left(\cos z-\frac{\sin z}{z}\right),$
27: 22.11 Fourier and Hyperbolic Series
22.11.1 $\operatorname{sn}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n% +\frac{1}{2}}\sin\left((2n+1)\zeta\right)}{1-q^{2n+1}},$
22.11.2 $\operatorname{cn}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n% +\frac{1}{2}}\cos\left((2n+1)\zeta\right)}{1+q^{2n+1}},$
22.11.3 $\operatorname{dn}\left(z,k\right)=\frac{\pi}{2K}+\frac{2\pi}{K}\sum_{n=1}^{% \infty}\frac{q^{n}\cos\left(2n\zeta\right)}{1+q^{2n}}.$
22.11.4 $\operatorname{cd}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{(-1)% ^{n}q^{n+\frac{1}{2}}\cos\left((2n+1)\zeta\right)}{1-q^{2n+1}},$
22.11.5 $\operatorname{sd}\left(z,k\right)=\frac{2\pi}{Kkk^{\prime}}\sum_{n=0}^{\infty}% \frac{(-1)^{n}q^{n+\frac{1}{2}}\sin\left((2n+1)\zeta\right)}{1+q^{2n+1}},$
28: 10.34 Analytic Continuation
10.34.1 $I_{\nu}\left(ze^{m\pi i}\right)=e^{m\nu\pi i}I_{\nu}\left(z\right),$
10.34.2 $K_{\nu}\left(ze^{m\pi i}\right)=e^{-m\nu\pi i}K_{\nu}\left(z\right)-\pi i\sin% \left(m\nu\pi\right)\csc\left(\nu\pi\right)I_{\nu}\left(z\right).$
10.34.4 $K_{\nu}\left(ze^{m\pi i}\right)=\csc\left(\nu\pi\right)\left(\pm\sin\left(m\nu% \pi\right)K_{\nu}\left(ze^{\pm\pi i}\right)\mp\sin\left((m\mp 1)\nu\pi\right)K% _{\nu}\left(z\right)\right).$
10.34.5 $K_{n}\left(ze^{m\pi i}\right)=(-1)^{mn}K_{n}\left(z\right)+(-1)^{n(m-1)-1}m\pi iI% _{n}\left(z\right),$
10.34.6 $K_{n}\left(ze^{m\pi i}\right)=\pm(-1)^{n(m-1)}mK_{n}\left(ze^{\pm\pi i}\right)% \mp(-1)^{nm}(m\mp 1)K_{n}\left(z\right).$
29: 11.7 Integrals and Sums
11.7.2 $\int z^{-\nu}\mathbf{H}_{\nu+1}\left(z\right)\mathrm{d}z=-z^{-\nu}\mathbf{H}_{% \nu}\left(z\right)+\frac{2^{-\nu}z}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}% \right)},$
11.7.4 $\int z^{-\nu}\mathbf{L}_{\nu+1}\left(z\right)\mathrm{d}z=z^{-\nu}\mathbf{L}_{% \nu}\left(z\right)-\frac{2^{-\nu}z}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}% \right)}.$
11.7.6 $f_{\nu+1}(z)=(2\nu+1)f_{\nu}(z)-z^{\nu+1}\mathbf{H}_{\nu}\left(z\right)+\frac{% (\tfrac{1}{2}z^{2})^{\nu+1}}{(\nu+1)\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}% \right)},$ $\Re\nu>-1$.
11.7.9 $\int_{0}^{\infty}\mathbf{H}_{\nu}\left(t\right)\mathrm{d}t=-\cot\left(\tfrac{1% }{2}\pi\nu\right),$ $-2<\Re\nu<0$,
11.7.10 $\int_{0}^{\infty}t^{-\nu-1}\mathbf{H}_{\nu}\left(t\right)\mathrm{d}t=\frac{\pi% }{2^{\nu+1}\Gamma\left(\nu+1\right)},$ $\Re\nu>-\tfrac{3}{2}$,