with respect to modulus

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1: 19.13 Integrals of Elliptic Integrals

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§19.13(i) Integration with Respect to the Modulus

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2: 22.13 Derivatives and Differential Equations

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22.13.2 $${\left(\frac{d}{dz}\mathrm{cn}(z,k)\right)}^2=\left(1-{\mathrm{cn}}^2(z,k)\right)\left(k_{}^{\prime }{}_{}{}^2+k^2{\mathrm{cn}}^2(z,k)\right),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E2

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See also:

Annotations for §22.13(ii), §22.13 and Ch.22

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22.13.3 $${\left(\frac{d}{dz}\mathrm{dn}(z,k)\right)}^2=\left(1-{\mathrm{dn}}^2(z,k)\right)\left({\mathrm{dn}}^2(z,k)-k_{}^{\prime }{}_{}{}^2\right).$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(ii), §22.13 and Ch.22

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22.13.5 $${\left(\frac{d}{dz}\mathrm{sd}(z,k)\right)}^2=\left(1-k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^2(z,k)\right)\left(1+k^2{\mathrm{sd}}^2(z,k)\right),$$

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Symbols:

$\mathrm{sd}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(ii), §22.13 and Ch.22

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22.13.6 $${\left(\frac{d}{dz}\mathrm{nd}(z,k)\right)}^2=\left({\mathrm{nd}}^2(z,k)-1\right)\left(1-k_{}^{\prime }{}_{}{}^2{\mathrm{nd}}^2(z,k)\right),$$

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Symbols:

$\mathrm{nd}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(ii), §22.13 and Ch.22

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22.13.8 $${\left(\frac{d}{dz}\mathrm{nc}(z,k)\right)}^2=\left(k^2+k_{}^{\prime }{}_{}{}^2{\mathrm{nc}}^2(z,k)\right)\left({\mathrm{nc}}^2(z,k)-1\right),$$

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Symbols:

$\mathrm{nc}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(ii), §22.13 and Ch.22

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3: 10.68 Modulus and Phase Functions

§10.68 Modulus and Phase Functions

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§10.68(i) Definitions

… ►where $M_{\nu }\left(x\right)(>0)$, $N_{\nu }\left(x\right)(>0)$, ${\theta }_{\nu }\left(x\right)$, and ${\varphi }_{\nu }\left(x\right)$ are continuous real functions of $x$ and $\nu$, with the branches of ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ chosen to satisfy (10.68.18) and (10.68.21) as $x\to \mathrm{\infty }$. … ►Equations (10.68.8)–(10.68.14) also hold with the symbols $\mathrm{ber}$, $\mathrm{bei}$, $M$, and $\theta$ replaced throughout by $\ker$, $\mathrm{kei}$, $N$, and $\varphi$, respectively. … ►10)), and ${lim}_{x\to \mathrm{\infty }}({\varphi }_1\left(x\right)+(x/\sqrt{2}))=-\frac{5}{8}\pi$ (Eqs. …

4: 19.4 Derivatives and Differential Equations

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19.4.3 $$\frac{d^{2}E\left(k\right)}{{dk}^2}=-\frac{1}{k}\frac{dK\left(k\right)}{dk}=\frac{k_{}^{\prime }{}_{}{}^{2}K\left(k\right)-E\left(k\right)}{k^{2}k_{}^{\prime }{}_{}{}^2},$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/19.4.E3

Encodings:

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See also:

Annotations for §19.4(i), §19.4 and Ch.19

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19.4.4 $$\frac{\partial \mathrm{\Pi }({\alpha }^2,k)}{\partial k}=\frac{k}{k_{}^{\prime }{}_{}{}^2(k^2-{\alpha }^2)}(E\left(k\right)-k_{}^{\prime }{}_{}{}^2\mathrm{\Pi }({\alpha }^2,k)).$$

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Symbols:

$E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: real or complex modulus, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter

Permalink:

http://dlmf.nist.gov/19.4.E4

Encodings:

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See also:

Annotations for §19.4(i), §19.4 and Ch.19

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19.4.5 $$\frac{\partial F(\varphi ,k)}{\partial k}=\frac{E(\varphi ,k)-k_{}^{\prime }{}_{}{}^{2}F(\varphi ,k)}{kk_{}^{\prime }{}_{}{}^2}-\frac{k\sin \varphi \cos \varphi }{k_{}^{\prime }{}_{}{}^2\sqrt{1-k^2{\sin }^2\varphi }},$$

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Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/19.4.E5

Encodings:

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See also:

Annotations for §19.4(i), §19.4 and Ch.19

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19.4.6 $$\frac{\partial E(\varphi ,k)}{\partial k}=\frac{E(\varphi ,k)-F(\varphi ,k)}{k},$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.30(i)

Permalink:

http://dlmf.nist.gov/19.4.E6

Encodings:

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See also:

Annotations for §19.4(i), §19.4 and Ch.19

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19.4.7 $$\frac{\partial \mathrm{\Pi }(\varphi ,{\alpha }^2,k)}{\partial k}=\frac{k}{k_{}^{\prime }{}_{}{}^2(k^2-{\alpha }^2)}\left(E(\varphi ,k)-k_{}^{\prime }{}_{}{}^2\mathrm{\Pi }(\varphi ,{\alpha }^2,k)-\frac{k^2\sin \varphi \cos \varphi }{\sqrt{1-k^2{\sin }^2\varphi }}\right).$$

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Symbols:

$\cos z$: cosine function, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter

Permalink:

http://dlmf.nist.gov/19.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

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5: 7.23 Tables

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Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\mathrm{erf}z$, $x\in [0,5]$, $y=0.5(.5)3$, 7D and 8D, respectively; the real and imaginary parts of ${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $(1/\sqrt{\pi }){\mathrm{e}}^{\mp \mathrm{i}(x^2+(\pi /4))}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $x=0(.5)20(1)25$, 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

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6: 3.11 Approximation Techniques

… ►The functions ${\varphi }_k(x)$ are orthogonal on the set $x_0,x_1,\mathrm{\dots },x_{n-1}$, $x_j=2\pi j/n$, with respect to the weight function $w(x)=1$, in the sense that …

7: 8.1 Special Notation

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$x$	real variable.
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$\psi \left(z\right)$	${\mathrm{\Gamma }}^{\prime }\left(z\right)/\mathrm{\Gamma }\left(z\right)$.

►Unless otherwise indicated, primes denote derivatives with respect to the argument. … ►Alternative notations include: Prym’s functions $P_z(a)=\gamma (a,z)$, $Q_z(a)=\mathrm{\Gamma }(a,z)$, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); $(a,z)!=\gamma (a+1,z)$, $[a,z]!=\mathrm{\Gamma }(a+1,z)$, Dingle (1973); $B(a,b,x)={\mathrm{B}}_x(a,b)$, $I(a,b,x)=I_x(a,b)$, Magnus et al. (1966); $\mathrm{Si}(a,x)\to \mathrm{Si}(1-a,x)$, $\mathrm{Ci}(a,x)\to \mathrm{Ci}(1-a,x)$, Luke (1975).

8: 15.1 Special Notation

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$x$	real variable.
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$\psi \left(z\right)$	${\mathrm{\Gamma }}^{\prime }\left(z\right)/\mathrm{\Gamma }\left(z\right)$.

►Unless indicated otherwise primes denote derivatives with respect to the variable. …

9: 13.8 Asymptotic Approximations for Large Parameters

… ►For asymptotic approximations to $M(a,b,x)$ and $U(a,b,x)$ as $a\to -\mathrm{\infty }$ that hold uniformly with respect to $x\in (0,\mathrm{\infty })$ and bounded positive values of $(b-1)/\left|a\right|$, combine (13.14.4), (13.14.5) with §§13.21(ii), 13.21(iii). …

10: 7.7 Integral Representations

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7.7.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-a{t}^2+2\mathrm{i}zt}dt=\frac{1}{2}\sqrt{\frac{\pi }{a}}{\mathrm{e}}^{-z^2/a}+\frac{\mathrm{i}}{\sqrt{a}}F\left(\frac{z}{\sqrt{a}}\right),$$ $\mathrm{\Re }a>0$.

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Symbols:

$F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

7.4.6 7.4.7 (in different notation)

Referenced by:

§7.14(i), §7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E3

Encodings:

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See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.6 $${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-(a{t}^2+2bt+c)}dt=\frac{1}{2}\sqrt{\frac{\pi }{a}}{\mathrm{e}}^{(b^2-ac)/a}\mathrm{erfc}\left(\sqrt{a}x+\frac{b}{\sqrt{a}}\right),$$ $\mathrm{\Re }a>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

A&S Ref:

7.4.32 (in different form)

Referenced by:

§7.14(i), §7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E6

Encodings:

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See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.7 $${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-a^2{t}^2-(b^2/t^2)}dt=\frac{\sqrt{\pi }}{4a}\left({\mathrm{e}}^{2ab}\mathrm{erfc}\left(ax+(b/x)\right)+{\mathrm{e}}^{-2ab}\mathrm{erfc}\left(ax-(b/x)\right)\right),$$ $x>0$, .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $x$: real variable

A&S Ref:

7.4.33 (in different form)

Referenced by:

§7.7(i), §7.8

Permalink:

http://dlmf.nist.gov/7.7.E7

Encodings:

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See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.8 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-a^2{t}^2-(b^2/t^2)}dt=\frac{\sqrt{\pi }}{2a}{\mathrm{e}}^{-2ab},$$ , .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{ph}$: phase

A&S Ref:

7.4.3 (in different form)

Referenced by:

§7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.12 $$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)={\mathrm{e}}^{-\pi \mathrm{i}{z}^2/2}{\int }_z^{\mathrm{\infty }}{\mathrm{e}}^{\pi \mathrm{i}{t}^2/2}dt.$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable

Referenced by:

§7.7(ii)

Permalink:

http://dlmf.nist.gov/7.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(ii), §7.7 and Ch.7

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