finite

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11: 24.4 Basic Properties

… ►

§24.4(iv) Finite Expansions

…

12: 8.25 Methods of Computation

… ►Although the series expansions in §§8.7, 8.19(iv), and 8.21(vi) converge for all finite values of

z

, they are cumbersome to use when

|z|

is large owing to slowness of convergence and cancellation. …

13: 15.17 Mathematical Applications

… ►These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …

14: 25.15 Dirichlet $L$ -functions

… ►

25.15.3

L\left(s,\chi\right)=k^{-s}\sum_{r=1}^{k-1}\chi(r)\zeta\left(s,\frac{r}{k}% \right),

ⓘ

Symbols:: $L\left(\NVar{s},\NVar{\chi}\right)$ : Dirichlet $L$ -function, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $k$ : nonnegative integer, $s$ : complex variable and $\chi(n)$ : Dirichlet character
Keywords:: finite sum
Source:: Apostol (1976, (8), p. 255)
Notes:: In the original source the upper-index of the finite sum is $k$ . We use $k-1$ since $\chi(k)=0$ .
Referenced by:: (25.11.36), (25.11.36), §25.11(x), §25.15(i), Erratum (V1.0.23) for Equation (25.11.36)
Permalink:: http://dlmf.nist.gov/25.15.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §25.15(i), §25.15 and Ch.25

… ►

25.15.6

G(\chi)\equiv\sum_{r=1}^{k-1}\chi(r)e^{2\pi ir/k}.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\equiv$ : equals by definition, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $k$ : nonnegative integer and $\chi(n)$ : Dirichlet character
Keywords:: definition
Source:: Apostol (1976, p. 262)
Referenced by:: Erratum (V1.1.4) for Equation (25.15.6)
Permalink:: http://dlmf.nist.gov/25.15.E6
Encodings:: pMML, png, TeX
Clarification (effective with 1.1.4):: The upper-index of the finite sum which originally was $k$ , was replaced with $k-1$ since $\chi(k)=0$ .
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.15(i), §25.15 and Ch.25

… ►

25.15.10

L\left(0,\chi\right)=\begin{cases}\displaystyle-\frac{1}{k}\sum_{r=1}^{k-1}r% \chi(r),&\chi\neq\chi_{1},\\ 0,&\chi=\chi_{1}.\end{cases}

ⓘ

Symbols:: $L\left(\NVar{s},\NVar{\chi}\right)$ : Dirichlet $L$ -function, $k$ : nonnegative integer and $\chi(n)$ : Dirichlet character
Keywords:: zeros
Source:: Apostol (1976, Theorem 12.20, p. 268)
Referenced by:: Erratum (V1.1.4) for Equation (25.15.10)
Permalink:: http://dlmf.nist.gov/25.15.E10
Encodings:: pMML, png, TeX
Clarification (effective with 1.1.4):: The upper-index of the finite sum which originally was $k$ , was replaced with $k-1$ since $\chi(k)=0$ .
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.15(ii), §25.15 and Ch.25

15: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►is a polynomial of degree

n

, and hence a solution of (31.2.1) that is analytic at all three finite singularities

0,1,a

. …

16: 31.6 Path-Multiplicative Solutions

… ►This denotes a set of solutions of (31.2.1) with the property that if we pass around a simple closed contour in the

z

-plane that encircles

s_{1}

and

s_{2}

once in the positive sense, but not the remaining finite singularity, then the solution is multiplied by a constant factor

{\mathrm{e}}^{2\nu\pi i}

. …

17: 34.2 Definition: $\mathit{3j}$ Symbol

… ►When both conditions are satisfied the

\mathit{3j}

symbol can be expressed as the finite sum … ►where

{{}_{3}F_{2}}

is defined as in §16.2. ►For alternative expressions for the

\mathit{3j}

symbol, written either as a finite sum or as other terminating generalized hypergeometric series

{{}_{3}F_{2}}

of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).

18: 34.4 Definition: $\mathit{6j}$ Symbol

§34.4 Definition: $\mathit{6j}$ Symbol

… ►The

\mathit{6j}

symbol can be expressed as the finite sum … ►For alternative expressions for the

\mathit{6j}

symbol, written either as a finite sum or as other terminating generalized hypergeometric series

{{}_{4}F_{3}}

of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).

19: 31.14 General Fuchsian Equation

… ►The exponents at the finite singularities

a_{j}

are

\{0,{1-\gamma_{j}}\}

and those at

\infty

are

\{\alpha,\beta\}

, where … ►An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions). …

20: 23.20 Mathematical Applications

… ►Let

T

denote the set of points on

C

that are of finite order (that is, those points

P

for which there exists a positive integer

n

with

nP=o

), and let

I, K

be the sets of points with integer and rational coordinates, respectively. …Both

T

and

I

are finite sets. …To determine

T

, we make use of the fact that if

(x,y)\in T

then

y^{2}

must be a divisor of

\Delta

; hence there are only a finite number of possibilities for

y

. …The resulting points are then tested for finite order as follows. …If any of these quantities is zero, then the point has finite order. …