limiting forms as trigonometric functions

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31: 18.10 Integral Representations

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18.10.2 $$P_n\left(\cos \theta \right)=\frac{2^{\frac{1}{2}}}{\pi }{\int }_0^{\theta }\frac{\cos \left((n+\frac{1}{2})\varphi \right)}{{(\cos \varphi -\cos \theta )}^{\frac{1}{2}}}d\varphi ,$$ .

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

A&S Ref:

22.10.13

Referenced by:

§18.10(i)

Permalink:

http://dlmf.nist.gov/18.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(i), §18.10(i), §18.10 and Ch.18

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18.10.5 $$P_n\left(\cos \theta \right)=\frac{1}{\pi }{\int }_0^{\pi }{(\cos \theta +i\sin \theta \cos \varphi )}^{n}d\varphi .$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $n$: nonnegative integer

A&S Ref:

22.10.12

Referenced by:

§18.10(ii)

Permalink:

http://dlmf.nist.gov/18.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

… ►Table 18.10.1 gives contour integral representations of the form … ►

Laguerre

… ►For the Bessel function $J_{\nu }\left(z\right)$ see §10.2(ii). …

32: 1.5 Calculus of Two or More Variables

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Implicit Function Theorem

… ►Sufficient conditions for the limit to exist are that $f(x,y)$ is continuous, or piecewise continuous, on $R$. … ►If $D$ can be represented in both forms (1.5.30) and (1.5.33), and $f(x,y)$ is continuous on $D$, then … ►In the cases (1.5.30) and (1.5.33) they are defined by taking limits in the repeated integrals (1.5.32) and (1.5.34) in an analogous manner to (1.4.22)–(1.4.23). … ►In case of triple integrals the $(x,y,z)$ sets are of the form …

33: Errata

… ►We have also incorporated material on continuous $q$-Jacobi polynomials, and several new limit transitions. … ►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. … ►

A sentence and unnumbered equation

$$R_C(\gamma -\delta ,\gamma )=\frac{-1}{\sqrt{\delta }}\arctan \left(\frac{\sqrt{\delta }\sin \theta \sin \varphi \sin \psi }{{\alpha }^2-1-{\alpha }^2\cos \theta \cos \varphi \cos \psi }\right),$$

were added which indicate that care must be taken with the multivalued functions in (19.11.5). See (Cayley, 1961, pp. 103-106).

Suggested by Albert Groenenboom.

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22.19.2 $$\sin \left(\frac{1}{2}\theta (t)\right)=\sin \left(\frac{1}{2}\alpha \right)\mathrm{sn}(t+K,\sin \left(\frac{1}{2}\alpha \right))$$

Originally the first argument to the function $\mathrm{sn}$ was given incorrectly as $t$. The correct argument is $t+K$.

Reported 2014-03-05 by Svante Janson.

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Originally the limiting form for $\mathrm{sc}(z,k)$ in the last line of this table was incorrect ($\cosh z$, instead of $\sinh z$).

$\mathrm{sn}(z,k)$ $\to$	$\tanh z$	$\mathrm{cd}(z,k)$ $\to$	$1$	$\mathrm{dc}(z,k)$ $\to$	$1$	$\mathrm{ns}(z,k)$ $\to$	$\coth z$
$\mathrm{cn}(z,k)$ $\to$	$\mathrm{sech}z$	$\mathrm{sd}(z,k)$ $\to$	$\sinh z$	$\mathrm{nc}(z,k)$ $\to$	$\cosh z$	$\mathrm{ds}(z,k)$ $\to$	$\mathrm{csch}z$
$\mathrm{dn}(z,k)$ $\to$	$\mathrm{sech}z$	$\mathrm{nd}(z,k)$ $\to$	$\cosh z$	$\mathrm{sc}(z,k)$ $\to$	$\sinh z$	$\mathrm{cs}(z,k)$ $\to$	$\mathrm{csch}z$

Reported 2010-11-23.

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34: 8.19 Generalized Exponential Integral

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§8.19(i) Definition and Integral Representations

… ►In Figures 8.19.2–8.19.5, height corresponds to the absolute value of the function and color to the phase. … ►The right-hand sides are replaced by their limiting forms when $p=1,2,3,\mathrm{\dots }$. … ►

§8.19(vi) Relation to Confluent Hypergeometric Function

… ►where …

35: 28.2 Definitions and Basic Properties

… ►The standard form of Mathieu’s equation with parameters $(a,q)$ is …With $\zeta ={\sin }^{2}z$ we obtain the algebraic form of Mathieu’s equation …With $\zeta =\cos z$ we obtain another algebraic form: … ► $\cos \left(\pi \nu \right)$ is an entire function of $a,q^2$. … ►

§28.2(vi) Eigenfunctions

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36: 1.9 Calculus of a Complex Variable

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Continuity

►A function $f(z)$ is continuous at a point $z_0$ if $\underset{z\to z_0}{lim}f(z)=f(z_0)$. … ►A function $f(z)$ is complex differentiable at a point $z$ if the following limit exists: … ►or its limiting form, and is invariant under bilinear transformations. … ►Then both repeated limits equal $z$. …

37: 32.2 Differential Equations

… ►For arbitrary values of the parameters $\alpha$, $\beta$, $\gamma$, and $\delta$, the general solutions of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are transcendental, that is, they cannot be expressed in closed-form elementary functions. … ►

§32.2(iii) Alternative Forms

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§32.2(iv) Elliptic Form

► ${\text{P}}_{\text{VI}}$ can be written in the form … ►

§32.2(v) Symmetric Forms

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38: 18.18 Sums

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§18.18(i) Series Expansions of Arbitrary Functions

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Expansion of $L^2$ functions

… ►See (18.5.11) for the limit case $\lambda \to 0$ of (18.18.16). … ►See (18.17.45) and (18.17.49) for integrated forms of (18.18.22) and (18.18.23), respectively. … ►

Laguerre

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39: 13.23 Integrals

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§13.23(i) Laplace and Mellin Transforms

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§13.23(ii) Fourier Transforms

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§13.23(iii) Hankel Transforms

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§13.23(iv) Integral Transforms in terms of Whittaker Functions

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