# left-hand

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##### 2: 13.5 Continued Fractions
This continued fraction converges to the meromorphic function of $z$ on the left-hand side everywhere in $\mathbb{C}$. … This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector $|\operatorname{ph}{z}|<\pi$. …
##### 3: 13.17 Continued Fractions
This continued fraction converges to the meromorphic function of $z$ on the left-hand side for all $z\in\mathbb{C}$. … This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector $|\operatorname{ph}{z}|<\pi$. …
##### 4: 5.10 Continued Fractions
5.10.1 $\operatorname{Ln}\Gamma\left(z\right)+z-\left(z-\tfrac{1}{2}\right)\ln z-% \tfrac{1}{2}\ln\left(2\pi\right)=\cfrac{a_{0}}{z+\cfrac{a_{1}}{z+\cfrac{a_{2}}% {z+\cfrac{a_{3}}{z+\cfrac{a_{4}}{z+\cfrac{a_{5}}{z+}}}}}}\cdots,$
##### 5: 10.70 Zeros
In the case $\nu=0$, numerical tabulations (Abramowitz and Stegun (1964, Table 9.12)) indicate that each of (10.70.2) corresponds to the $m$th zero of the function on the left-hand side. …
##### 6: 19.10 Relations to Other Functions
In each case when $y=1$, the quantity multiplying $R_{C}$ supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0. …
##### 7: 15.6 Integral Representations
15.6.1 $\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(b\right)\Gamma\left(c-b% \right)}\int_{0}^{1}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\,\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $\Re c>\Re b>0$.
15.6.2 $\mathbf{F}\left(a,b;c;z\right)=\frac{\Gamma\left(1+b-c\right)}{2\pi\mathrm{i}% \Gamma\left(b\right)}\int_{0}^{(1+)}\frac{t^{b-1}(t-1)^{c-b-1}}{(1-zt)^{a}}\,% \mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $c-b\neq 1,2,3,\dots$, $\Re b>0$.
15.6.3 $\mathbf{F}\left(a,b;c;z\right)={\mathrm{e}}^{-b\pi\mathrm{i}}\frac{\Gamma\left% (1-b\right)}{2\pi\mathrm{i}\Gamma\left(c-b\right)}\int_{\infty}^{(0+)}\frac{t^% {b-1}(t+1)^{a-c}}{(t-zt+1)^{a}}\,\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $b\neq 1,2,3,\dots$, $\Re\left(c-b\right)>0$.
15.6.4 $\mathbf{F}\left(a,b;c;z\right)={\mathrm{e}}^{-b\pi\mathrm{i}}\frac{\Gamma\left% (1-b\right)}{2\pi\mathrm{i}\Gamma\left(c-b\right)}\int_{1}^{(0+)}\frac{t^{b-1}% (1-t)^{c-b-1}}{(1-zt)^{a}}\,\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $b\neq 1,2,3,\dots$, $\Re\left(c-b\right)>0$.
15.6.8 $\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(c-d\right)}\int_{0}^{1}% \mathbf{F}\left(a,b;d;zt\right)t^{d-1}(1-t)^{c-d-1}\,\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $\Re c>\Re d>0$.
##### 8: 16.5 Integral Representations and Integrals
Then the integral converges when $p provided that $z\neq 0$, or when $p=q+1$ provided that $0<|z|<1$, and provides an integral representation of the left-hand side with these conditions. … In the case $p=q$ the left-hand side of (16.5.1) is an entire function, and the right-hand side supplies an integral representation valid when $|\operatorname{ph}\left(-z\right)|<\pi/2$. In the case $p=q+1$ the right-hand side of (16.5.1) supplies the analytic continuation of the left-hand side from the open unit disk to the sector $|\operatorname{ph}\left(1-z\right)|<\pi$; compare §16.2(iii). Lastly, when $p>q+1$ the right-hand side of (16.5.1) can be regarded as the definition of the (customarily undefined) left-hand side. In this event, the formal power-series expansion of the left-hand side (obtained from (16.2.1)) is the asymptotic expansion of the right-hand side as $z\to 0$ in the sector $|\operatorname{ph}\left(-z\right)|\leq(p+1-q-\delta)\pi/2$, where $\delta$ is an arbitrary small positive constant. …
##### 10: 24.19 Methods of Computation
• Buhler et al. (1992) uses the expansion

and computes inverses modulo $p$ of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).