# relation to line broadening function

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##### 1: 23.15 Definitions
###### §23.15 Definitions
A modular function $f(\tau)$ is a function of $\tau$ that is meromorphic in the half-plane $\Im\tau>0$, and has the property that for all $\mathcal{A}\in\mbox{SL}(2,\mathbb{Z})$, or for all $\mathcal{A}$ belonging to a subgroup of SL$(2,\mathbb{Z})$, …(Some references refer to $2\ell$ as the level). …If, in addition, $f(\tau)\to 0$ as $q\to 0$, then $f(\tau)$ is called a cusp form. …
##### 2: 14.19 Toroidal (or Ring) Functions
###### §14.19(i) Introduction
This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates $(\eta,\theta,\phi)$, which are related to Cartesian coordinates $(x,y,z)$ by …
##### 3: 9.12 Scorer Functions
###### §9.12 Scorer Functions
$-\operatorname{Gi}\left(x\right)$ is a numerically satisfactory companion to the complementary functions $\operatorname{Ai}\left(x\right)$ and $\operatorname{Bi}\left(x\right)$ on the interval $0\leq x<\infty$. $\operatorname{Hi}\left(x\right)$ is a numerically satisfactory companion to $\operatorname{Ai}\left(x\right)$ and $\operatorname{Bi}\left(x\right)$ on the interval $-\infty. … As $z\to\infty$, and with $\delta$ denoting an arbitrary small positive constant, …
##### 4: 9.1 Special Notation
(For other notation see Notation for the Special Functions.)
 $k$ nonnegative integer, except in §9.9(iii). … derivatives with respect to argument.
The main functions treated in this chapter are the Airy functions $\operatorname{Ai}\left(z\right)$ and $\operatorname{Bi}\left(z\right)$, and the Scorer functions $\operatorname{Gi}(z)$ and $\operatorname{Hi}(z)$ (also known as inhomogeneous Airy functions). Other notations that have been used are as follows: $\operatorname{Ai}\left(-x\right)$ and $\operatorname{Bi}\left(-x\right)$ for $\operatorname{Ai}\left(x\right)$ and $\operatorname{Bi}\left(x\right)$ (Jeffreys (1928), later changed to $\operatorname{Ai}\left(x\right)$ and $\operatorname{Bi}\left(x\right)$); $U(x)=\sqrt{\pi}\operatorname{Bi}\left(x\right)$, $V(x)=\sqrt{\pi}\operatorname{Ai}\left(x\right)$ (Fock (1945)); $A(x)=3^{-\ifrac{1}{3}}\pi\operatorname{Ai}\left(-3^{-\ifrac{1}{3}}x\right)$ (Szegő (1967, §1.81)); $e_{0}(x)=\pi\operatorname{Hi}(-x)$, $\widetilde{e}_{0}(x)=-\pi\operatorname{Gi}(-x)$ (Tumarkin (1959)).
##### 5: 20.2 Definitions and Periodic Properties
###### §20.2(i) Fourier Series
Corresponding expansions for $\theta_{j}'\left(z\middle|\tau\right)$, $j=1,2,3,4$, can be found by differentiating (20.2.1)–(20.2.4) with respect to $z$.
###### §20.2(ii) Periodicity and Quasi-Periodicity
The theta functions are quasi-periodic on the lattice: …
##### 6: 15.2 Definitions and Analytical Properties
###### §15.2(i) Gauss Series
The branch obtained by introducing a cut from $1$ to $+\infty$ on the real $z$-axis, that is, the branch in the sector $|\operatorname{ph}\left(1-z\right)|\leq\pi$, is the principal branch (or principal value) of $F\left(a,b;c;z\right)$. … Because of the analytic properties with respect to $a$, $b$, and $c$, it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. … which sometimes needs to be used in §15.4. … In that case we are using interpretation (15.2.6) since with interpretation (15.2.5) we would obtain that $F\left(-m,b;-m;z\right)$ is equal to the first $m+1$ terms of the Maclaurin series for $(1-z)^{-b}$.
##### 7: 5.12 Beta Function
###### Euler’s Beta Integral
5.12.2 $\int_{0}^{\pi/2}{\sin}^{2a-1}\theta{\cos}^{2b-1}\theta\,\mathrm{d}\theta=% \tfrac{1}{2}\mathrm{B}\left(a,b\right).$
###### Pochhammer’s Integral
where the contour starts from an arbitrary point $P$ in the interval $(0,1)$, circles $1$ and then $0$ in the positive sense, circles $1$ and then $0$ in the negative sense, and returns to $P$. …
##### 8: 5.15 Polygamma Functions
###### §5.15 Polygamma Functions
The functions $\psi^{(n)}\left(z\right)$, $n=1,2,\dots$, are called the polygamma functions. …Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … As $z\to\infty$ in $|\operatorname{ph}z|\leq\pi-\delta$ …For $B_{2k}$ see §24.2(i). …
##### 9: 16.13 Appell Functions
###### §16.13 Appell Functions
The following four functions of two real or complex variables $x$ and $y$ cannot be expressed as a product of two ${{}_{2}F_{1}}$ functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1):
16.13.1 ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=\sum_{m,n=0}^{% \infty}\frac{{\left(\alpha\right)_{m+n}}{\left(\beta\right)_{m}}{\left(\beta^{% \prime}\right)_{n}}}{{\left(\gamma\right)_{m+n}}m!n!}x^{m}y^{n},$ $\max\left(|x|,|y|\right)<1$,
16.13.4 ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)=\sum_{m,n=0}^{% \infty}\frac{{\left(\alpha\right)_{m+n}}{\left(\beta\right)_{m+n}}}{{\left(% \gamma\right)_{m}}{\left(\gamma^{\prime}\right)_{n}}m!n!}x^{m}y^{n},$ $\sqrt{|x|}+\sqrt{|y|}<1$.
##### 10: 14.20 Conical (or Mehler) Functions
###### §14.20(iii) Behavior as $x\to 1$
As $\tau\to\infty$, … As $\mu\to\infty$, …