# implicit function theorem

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##### 1: 15.2 Definitions and Analytical Properties
###### §15.2(i) Gauss Series
The hypergeometric function $F\left(a,b;c;z\right)$ is defined by the Gauss series … … On the circle of convergence, $|z|=1$, the Gauss series: …
##### 2: 9.1 Special Notation
(For other notation see Notation for the Special Functions.)
 $k$ nonnegative integer, except in §9.9(iii). …
The main functions treated in this chapter are the Airy functions $\mathrm{Ai}\left(z\right)$ and $\mathrm{Bi}\left(z\right)$, and the Scorer functions $\mathrm{Gi}(z)$ and $\mathrm{Hi}(z)$ (also known as inhomogeneous Airy functions). Other notations that have been used are as follows: $\mathrm{Ai}\left(-x\right)$ and $\mathrm{Bi}\left(-x\right)$ for $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ (Jeffreys (1928), later changed to $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$); $U(x)=\sqrt{\pi}\mathrm{Bi}\left(x\right)$, $V(x)=\sqrt{\pi}\mathrm{Ai}\left(x\right)$ (Fock (1945)); $A(x)=3^{-\ifrac{1}{3}}\pi\mathrm{Ai}\left(-3^{-\ifrac{1}{3}}x\right)$ (Szegő (1967, §1.81)); $e_{0}(x)=\pi\mathrm{Hi}(-x)$, $\widetilde{e}_{0}(x)=-\pi\mathrm{Gi}(-x)$ (Tumarkin (1959)).
##### 4: 31.1 Special Notation
(For other notation see Notation for the Special Functions.)
 $x$, $y$ real variables. …
The main functions treated in this chapter are $\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$, $(s_{2},)\mathit{Hf}_{s_{1}}^{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)$, $(s_{1},s_{2})\mathit{Hf}_{m}^{\nu}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)$, and the polynomial $\mathit{Hp}_{n,m}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)$. …Sometimes the parameters are suppressed.
##### 5: 5.15 Polygamma Functions
###### §5.15 Polygamma Functions
The functions $\psi^{(n)}\left(z\right)$, $n=1,2,\dots$, are called the polygamma functions. In particular, $\psi'\left(z\right)$ is the trigamma function; $\psi''$, $\psi^{(3)}$, $\psi^{(4)}$ are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … For $B_{2k}$ see §24.2(i). …
##### 6: 5.2 Definitions
###### Euler’s Integral
5.2.1 $\Gamma\left(z\right)=\int_{0}^{\infty}e^{-t}t^{z-1}\mathrm{d}t,$ $\Re z>0$.
It is a meromorphic function with no zeros, and with simple poles of residue $(-1)^{n}/n!$ at $z=-n$. …
5.2.2 $\psi\left(z\right)=\Gamma'\left(z\right)/\Gamma\left(z\right),$ $z\neq 0,-1,-2,\dots$.
where …
##### 10: 20.2 Definitions and Periodic Properties
###### §20.2(ii) Periodicity and Quasi-Periodicity
The theta functions are quasi-periodic on the lattice: …