relation to line broadening function

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21: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

… ►The Riemann zeta function is a special case: … ►For other series expansions similar to (25.11.10) see Coffey (2008). … ►When

a=1

, (25.11.35) reduces to (25.2.3). … ►uniformly with respect to bounded nonnegative values of

\alpha

. …

22: 1.10 Functions of a Complex Variable

… ►Analytic continuation is a powerful aid in establishing transformations or functional equations for complex variables, because it enables the problem to be reduced to: (a) deriving the transformation (or functional equation) with real variables; followed by (b) finding the domain on which the transformed function is analytic. … ►then the product

\prod^{\infty}_{n=1}(1+a_{n}(z))

converges uniformly to an analytic function

p(z)

D

, and

p(z)=0

only when at least one of the factors

1+a_{n}(z)

is zero in

D

. … ►

§1.10(x) Infinite Partial Fractions

… ►

§1.10(xi) Generating Functions

… ►The recurrence relation for

C^{(\lambda)}_{n}\left(x\right)

in §18.9(i) follows from

(1-2xz+z^{2})\frac{\partial}{\partial z}F(x,\lambda;z)=2\lambda(x-z)F(x,% \lambda;z)

, and the contour integral representation for

C^{(\lambda)}_{n}\left(x\right)

in §18.10(iii) is just (1.10.27).

23: 23.2 Definitions and Periodic Properties

… ►

§23.2(i) Lattices

… ►

§23.2(ii) Weierstrass Elliptic Functions

… ►When

z\notin\mathbb{L}

the functions are related by … ►When it is important to display the lattice with the functions they are denoted by

\wp\left(z|\mathbb{L}\right)

\zeta\left(z|\mathbb{L}\right)

, and

\sigma\left(z|\mathbb{L}\right)

, respectively. …

24: 4.23 Inverse Trigonometric Functions

§4.23 Inverse Trigonometric Functions

… ►The principal values (or principal branches) of the inverse sine, cosine, and tangent are obtained by introducing cuts in the

z

-plane as indicated in Figures 4.23.1(i) and 4.23.1(ii), and requiring the integration paths in (4.23.1)–(4.23.3) not to cross these cuts. … ►the upper/lower signs corresponding to the upper/lower sides. … ►the upper/lower sign corresponding to the right/left side. … ►Care needs to be taken on the cuts, for example, if

0<x<\infty

then

1/(x+i0)=(1/x)-i0

. …

25: 4.37 Inverse Hyperbolic Functions

§4.37 Inverse Hyperbolic Functions

… ►the upper/lower signs corresponding to the right/left sides. … ►the upper/lower sign corresponding to the upper/lower side. … ►the upper/lower sign corresponding to the upper/lower side. … ►Table 4.30.1 can also be used to find interrelations between inverse hyperbolic functions. …

26: 17.1 Special Notation

§17.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►The main functions treated in this chapter are the basic hypergeometric (or

q

-hypergeometric) function

{{}_{r}\phi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)

, the bilateral basic hypergeometric (or bilateral

q

-hypergeometric) function

{{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)

, and the

q

-analogs of the Appell functions

\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)

\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)

\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)

, and

\Phi^{(4)}\left(a,b;c,c^{\prime};q;x,y\right)

. ►Another function notation used is the “idem” function: …

27: 16.2 Definition and Analytic Properties

… ►

§16.2(i) Generalized Hypergeometric Series

… ►If none of the

a_{j}

is a nonpositive integer, then the radius of convergence of the series (16.2.1) is

1

, and outside the open disk

|z|<1

the generalized hypergeometric function is defined by analytic continuation with respect to

z

. The branch obtained by introducing a cut from

1

+\infty

on the real axis, that is, the branch in the sector

|\operatorname{ph}\left(1-z\right)|\leq\pi

, is the principal branch (or principal value) of

{{}_{q+1}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)

; compare §4.2(i). … ►can be used to interchange

p

and

q

. … ►

§16.2(v) Behavior with Respect to Parameters

…

28: 30.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►These notations are similar to those used in Arscott (1964b) and Erdélyi et al. (1955). Meixner and Schäfke (1954) use

\mathrm{ps}

\mathrm{qs}

\mathrm{Ps}

\mathrm{Qs}

for

\mathsf{Ps}

\mathsf{Qs}

\mathit{Ps}

\mathit{Qs}

, respectively. ►

Other Notations

…

29: 14.1 Special Notation

§14.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise. ►The main functions treated in this chapter are the Legendre functions

\mathsf{P}_{\nu}\left(x\right)

\mathsf{Q}_{\nu}\left(x\right)

P_{\nu}\left(z\right)

Q_{\nu}\left(z\right)

; Ferrers functions

\mathsf{P}^{\mu}_{\nu}\left(x\right)

\mathsf{Q}^{\mu}_{\nu}\left(x\right)

(also known as the Legendre functions on the cut); associated Legendre functions

P^{\mu}_{\nu}\left(z\right)

Q^{\mu}_{\nu}\left(z\right)

\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)

; conical functions

\mathsf{P}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

\mathsf{Q}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

\widehat{\mathsf{Q}}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

P^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

Q^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

(also known as Mehler functions). …

30: 16.17 Definition

§16.17 Definition

… ►

(i)

$L$ goes from $-\mathrm{i}\infty$ to $\mathrm{i}\infty$ . The integral converges if $p+q<2(m+n)$ and $|\operatorname{ph}z|<(m+n-\frac{1}{2}(p+q))\pi$ .

►

(ii)

$L$ is a loop that starts at infinity on a line parallel to the positive real axis, encircles the poles of the $\Gamma\left(b_{\ell}-s\right)$ once in the negative sense and returns to infinity on another line parallel to the positive real axis. The integral converges for all $z$ ( $\neq 0$ ) if $p<q$ , and for $0<|z|<1$ if $p=q\geq 1$ .

►

(iii)

$L$ is a loop that starts at infinity on a line parallel to the negative real axis, encircles the poles of the $\Gamma\left(1-a_{\ell}+s\right)$ once in the positive sense and returns to infinity on another line parallel to the negative real axis. The integral converges for all $z$ if $p>q$ , and for $|z|>1$ if $p=q\geq 1$ .

… ►Then …