Legendre%0Aelliptic%20integrals

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1: 14.1 Special Notation

§14.1 Special Notation

… ►The main functions treated in this chapter are the Legendre functions ${𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }\left(x\right)$, $P_{\nu }\left(z\right)$, $Q_{\nu }\left(z\right)$; Ferrers functions ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$; conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ (also known as Mehler functions). … ►Magnus et al. (1966) denotes ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$, $P_{\nu }^{\mu }\left(z\right)$, and $Q_{\nu }^{\mu }\left(z\right)$ by $P_{\nu }^{\mu }(x)$, $Q_{\nu }^{\mu }(x)$, ${𝔓}_{\nu }^{\mu }(z)$, and ${𝔔}_{\nu }^{\mu }(z)$, respectively. Hobson (1931) denotes both ${𝖯}_{\nu }^{\mu }\left(x\right)$ and $P_{\nu }^{\mu }\left(x\right)$ by $P_{\nu }^{\mu }\left(x\right)$; similarly for ${𝖰}_{\nu }^{\mu }\left(x\right)$ and $Q_{\nu }^{\mu }\left(x\right)$.

2: 18.3 Definitions

§18.3 Definitions

… ►This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. … ►Explicit power series for Chebyshev, Legendre, Laguerre, and Hermite polynomials for $n=0,1,\mathrm{\dots },6$ are given in §18.5(iv). … ►

Legendre

►Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …

3: 19.2 Definitions

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§19.2(i) General Elliptic Integrals

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§19.2(ii) Legendre’s Integrals

… ►Legendre’s complementary complete elliptic integrals are defined via … ►

§19.2(iii) Bulirsch’s Integrals

… ►Lastly, corresponding to Legendre’s incomplete integral of the third kind we have …

4: 1.14 Integral Transforms

§1.14 Integral Transforms

… ►Sufficient conditions for the integral to converge are that $s$ is a positive real number, and $f(t)=O\left(t^{-\delta }\right)$ as $t\to \mathrm{\infty }$, where $\delta >0$. … ►If the integral converges, then it converges uniformly in any compact domain in the complex $s$-plane not containing any point of the interval $(-\mathrm{\infty },0]$. … ►If $f(t)$ is absolutely integrable on $[0,R]$ for every finite $R$, and the integral (1.14.47) converges, then … ►If $f(t)$ is piecewise continuous on $[0,\mathrm{\infty })$ and the integral (1.14.47) converges, then …

5: 8.19 Generalized Exponential Integral

§8.19 Generalized Exponential Integral

… ►Unless $p$ is a nonpositive integer, $E_p\left(z\right)$ has a branch point at $z=0$. For $z\ne 0$ each branch of $E_p\left(z\right)$ is an entire function of $p$. … ►For $n=1,2,3,\mathrm{\dots }$ and $x>0$, … ►

§8.19(x) Integrals

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6: 6.2 Definitions and Interrelations

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§6.2(i) Exponential and Logarithmic Integrals

… ►As in the case of the logarithm (§4.2(i)) there is a cut along the interval $(-\mathrm{\infty },0]$ and the principal value is two-valued on $(-\mathrm{\infty },0)$. … ►In the next three equations $x>0$. …($\mathrm{Ei}\left(x\right)$ is undefined when $x=0$, or when $x$ is not real.) … ►

§6.2(ii) Sine and Cosine Integrals

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7: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

… ►From §§8.2(i) and 8.2(ii) it follows that each of the four functions $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ is a multivalued function of $z$ with branch point at $z=0$. Furthermore, $\mathrm{si}(a,z)$ and $\mathrm{ci}(a,z)$ are entire functions of $a$, and $\mathrm{Si}(a,z)$ and $\mathrm{Ci}(a,z)$ are meromorphic functions of $a$ with simple poles at $a=-1,-3,-5,\mathrm{\dots }$ and $a=0,-2,-4,\mathrm{\dots }$, respectively. … ►When $\mathrm{ph}z=0$ (and when $a\ne -1,-3,-5,\mathrm{\dots }$, in the case of $\mathrm{Si}(a,z)$, or $a\ne 0,-2,-4,\mathrm{\dots }$, in the case of $\mathrm{Ci}(a,z)$) the principal values of $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). … ►

8: 7.2 Definitions

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§7.2(ii) Dawson’s Integral

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§7.2(iii) Fresnel Integrals

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Values at Infinity

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§7.2(iv) Auxiliary Functions

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§7.2(v) Goodwin–Staton Integral

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9: 19.16 Definitions

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§19.16(i) Symmetric Integrals

… ►where $p$ ($\ne 0$) is a real or complex constant, and …In (19.16.1)–(19.16.2_5), $x,y,z\in ℂ\setminus (-\mathrm{\infty },0]$ except that one or more of $x,y,z$ may be 0 when the corresponding integral converges. … ►with the same conditions on $x$, $y$, $z$ as for (19.16.1), but now $z\ne 0$. … ►When one variable is 0 without destroying convergence, any one of (19.16.14)–(19.16.17) is said to be complete and can be written as an $R$-function with one less variable: …

10: 7.18 Repeated Integrals of the Complementary Error Function

§7.18 Repeated Integrals of the Complementary Error Function

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§7.18(i) Definition

… ►and for $n=0,1,2,\mathrm{\dots }$, … ► … ►

Hermite Polynomials

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