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1: 1.9 Calculus of a Complex Variable

… ►The cross ratio of $z_1,z_2,z_3,z_4\in ℂ\cup \{\mathrm{\infty }\}$ is defined by …

2: 10.28 Wronskians and Cross-Products

§10.28 Wronskians and Cross-Products

►

10.28.1 $$𝒲\left\{I_{\nu }\left(z\right),I_{-\nu }\left(z\right)\right\}=I_{\nu }\left(z\right)I_{-\nu -1}\left(z\right)-I_{\nu +1}\left(z\right)I_{-\nu }\left(z\right)=-2\sin \left(\nu \pi \right)/(\pi z),$$

ⓘ

Symbols:

$𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.14

Permalink:

http://dlmf.nist.gov/10.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.28 and Ch.10

…

3: 28.28 Integrals, Integral Representations, and Integral Equations

… ►

28.28.25 $$\frac{\sinh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\cos t{\mathrm{me}}_{\nu }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{m+1}\mathrm{i}h{\alpha }_{\nu ,m}^{(0)}{\mathrm{D}}_0(\nu ,\nu +2m+1,z),$$

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Symbols:

${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$

Permalink:

http://dlmf.nist.gov/28.28.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iii), §28.28 and Ch.28

►

28.28.26 $$\frac{\cosh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\sin t{\mathrm{me}}_{\nu }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{m+1}\mathrm{i}h{\alpha }_{\nu ,m}^{(1)}{\mathrm{D}}_0(\nu ,\nu +2m+1,z),$$

ⓘ

Symbols:

${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$

Permalink:

http://dlmf.nist.gov/28.28.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iii), §28.28 and Ch.28

… ►

28.28.29 $$\frac{\cosh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\sin t{\mathrm{me}}_{\nu }^{\prime }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{m+1}\mathrm{i}h{\alpha }_{\nu ,m}^{(0)}{\mathrm{D}}_1(\nu ,\nu +2m+1,z),$$

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Symbols:

${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$

Permalink:

http://dlmf.nist.gov/28.28.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iii), §28.28 and Ch.28

►

28.28.30 $$\frac{\sinh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\cos t{\mathrm{me}}_{\nu }^{\prime }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^m\mathrm{i}h{\alpha }_{\nu ,m}^{(1)}{\mathrm{D}}_1(\nu ,\nu +2m+1,z),$$

ⓘ

Symbols:

${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$

Permalink:

http://dlmf.nist.gov/28.28.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iii), §28.28 and Ch.28

… ►

28.28.49 $${\widehat{\alpha }}_{n,m}^{(c)}=\frac{1}{2\pi }{\int }_0^{2\pi }\cos t{\mathrm{ce}}_n(t,h^2){\mathrm{ce}}_m(t,h^2)dt={(-1)}^{p+1}\frac{2}{\mathrm{i}\pi }\frac{{\mathrm{ce}}_n(0,h^2){\mathrm{ce}}_m(0,h^2)}{h{\mathrm{Dc}}_0(n,m,0)}.$$

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Defines:

${\widehat{\alpha }}_{n,m}$ (locally)

Symbols:

${\mathrm{Dc}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $m$: integer, $h$: parameter and $n$: integer

Permalink:

http://dlmf.nist.gov/28.28.E49

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iv), §28.28 and Ch.28

…

4: 10.5 Wronskians and Cross-Products

§10.5 Wronskians and Cross-Products

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10.5.1 $$𝒲\left\{J_{\nu }\left(z\right),J_{-\nu }\left(z\right)\right\}=J_{\nu +1}\left(z\right)J_{-\nu }\left(z\right)+J_{\nu }\left(z\right)J_{-\nu -1}\left(z\right)=-2\sin \left(\nu \pi \right)/(\pi z),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.15

Permalink:

http://dlmf.nist.gov/10.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

►

10.5.2 $$𝒲\left\{J_{\nu }\left(z\right),Y_{\nu }\left(z\right)\right\}=J_{\nu +1}\left(z\right)Y_{\nu }\left(z\right)-J_{\nu }\left(z\right)Y_{\nu +1}\left(z\right)=2/(\pi z),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.16

Permalink:

http://dlmf.nist.gov/10.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

►

10.5.3 $$𝒲\left\{J_{\nu }\left(z\right),H_{\nu }^{(1)}\left(z\right)\right\}=J_{\nu +1}\left(z\right)H_{\nu }^{(1)}\left(z\right)-J_{\nu }\left(z\right)H_{\nu +1}^{(1)}\left(z\right)=2\mathrm{i}/(\pi z),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

►

10.5.4 $$𝒲\left\{J_{\nu }\left(z\right),H_{\nu }^{(2)}\left(z\right)\right\}=J_{\nu +1}\left(z\right)H_{\nu }^{(2)}\left(z\right)-J_{\nu }\left(z\right)H_{\nu +1}^{(2)}\left(z\right)=-2\mathrm{i}/(\pi z),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

…

5: 10.6 Recurrence Relations and Derivatives

… ►

10.6.10 $$p_{\nu }{s}_{\nu }-q_{\nu }{r}_{\nu }=4/({\pi }^{2}ab).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\nu$: complex parameter, $p_{\nu }$: cross-product, $q_{\nu }$: cross-product, $r_{\nu }$: cross-product and $s_{\nu }$: cross-product

A&S Ref:

9.1.34

Permalink:

http://dlmf.nist.gov/10.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.6(iii), §10.6 and Ch.10

6: 10.67 Asymptotic Expansions for Large Argument

… ►

10.67.1 $${\ker }_{\nu }x\sim {\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{x^k}\cos \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}+\frac{1}{8}\right)\pi \right),$$

ⓘ

Symbols:

${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.10.3, 9.10.5--9.10.7 (modified)

Referenced by:

§10.61(v), §10.67(ii), §10.68(iii)

Permalink:

http://dlmf.nist.gov/10.67.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(i), §10.67 and Ch.10

►

10.67.2 $${\mathrm{kei}}_{\nu }x\sim -{\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{x^k}\sin \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}+\frac{1}{8}\right)\pi \right).$$

ⓘ

Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.10.4, 9.10.5--9.10.7 (modified)

Referenced by:

§10.61(v)

Permalink:

http://dlmf.nist.gov/10.67.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(i), §10.67 and Ch.10

… ►

10.67.5 $${\ker }_{\nu }^{\prime }x\sim -{\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{b_k(\nu )}{x^k}\cos \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}-\frac{1}{8}\right)\pi \right),$$

ⓘ

Symbols:

${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $b_k(\nu )$: polynomial coefficient

A&S Ref:

§9.10 (modified)

Permalink:

http://dlmf.nist.gov/10.67.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(i), §10.67 and Ch.10

►

10.67.6 $${\mathrm{kei}}_{\nu }^{\prime }x\sim {\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{b_k(\nu )}{x^k}\sin \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}-\frac{1}{8}\right)\pi \right).$$

ⓘ

Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $b_k(\nu )$: polynomial coefficient

A&S Ref:

§9.10 (modified)

Permalink:

http://dlmf.nist.gov/10.67.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(i), §10.67 and Ch.10

… ►

§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu =0$

…

7: 14.2 Differential Equations

… ►

14.2.11 $$P_{\nu +1}^{\mu }\left(x\right)Q_{\nu }^{\mu }\left(x\right)-P_{\nu }^{\mu }\left(x\right)Q_{\nu +1}^{\mu }\left(x\right)={\mathrm{e}}^{\mu \pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\mathrm{\Gamma }\left(\nu -\mu +2\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.2(iv), §14.2 and Ch.14

8: 13.4 Integral Representations

… ►

13.4.4 $$U(a,b,z)=\frac{1}{\mathrm{\Gamma }\left(a\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{a-1}{(1+t)}^{b-a-1}dt,$$ $\mathrm{\Re }a>0$, ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

13.2.5

Referenced by:

§13.10(v), §13.29(iii), §7.11, (9.5.8)

Permalink:

http://dlmf.nist.gov/13.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

►

13.4.5 $$U(a,b,z)=\frac{z^{1-a}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(1+a-b\right)}{\int }_0^{\mathrm{\infty }}\frac{U(b-a,b,t){\mathrm{e}}^{-t}{t}^{a-1}}{t+z}dt,$$ , $\mathrm{\Re }a>\max (\mathrm{\Re }b-1,0)$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§13.10(vi), §13.4(i), (9.10.18)

Permalink:

http://dlmf.nist.gov/13.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

… ►

13.4.8 $$U(a,b,z)=z^{c-a}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{c-1}{}_2{}^{}𝐅_1^{}(a,a-b+1;c;-t)dt,$$ ,

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable and $c$: arbitrary

Permalink:

http://dlmf.nist.gov/13.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

… ►At the point where the contour crosses the interval $(1,\mathrm{\infty })$, $t^{-b}$ and the ${}_2{}^{}𝐅_1^{}$ function assume their principal values; compare §§15.1 and 15.2(i). … ►

13.4.13 $$𝐌(a,b,z)=\frac{z^{1-b}}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+,1+)}{\mathrm{e}}^{zt}{t}^{-b}{\left(1-\frac{1}{t}\right)}^{-a}dt,$$ .

ⓘ

Symbols:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: 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, $(a,b)$: open interval, $\mathrm{ph}$: phase and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.4.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(ii), §13.4 and Ch.13

…

9: 8.19 Generalized Exponential Integral

… ►When the path of integration excludes the origin and does not cross the negative real axis (8.19.2) defines the principal value of $E_p\left(z\right)$, and unless indicated otherwise in the DLMF principal values are assumed. … ►

8.19.3 $$E_p\left(z\right)={\int }_1^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}}{t^p}dt,$$ ,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable and $p$: parameter

Keywords:

Laplace transform, Mellin transform

A&S Ref:

5.1.4

Referenced by:

§8.19(iii)

Permalink:

http://dlmf.nist.gov/8.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(i), §8.19(i), §8.19 and Ch.8

►

8.19.4 $$E_p\left(z\right)=\frac{z^{p-1}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(p\right)}{\int }_0^{\mathrm{\infty }}\frac{t^{p-1}{\mathrm{e}}^{-zt}}{1+t}dt,$$ , $\mathrm{\Re }p>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable and $p$: parameter

Keywords:

Laplace transform, Mellin transform

Permalink:

http://dlmf.nist.gov/8.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(i), §8.19(i), §8.19 and Ch.8

… ►

8.19.17 $$E_p\left(z\right)={\mathrm{e}}^{-z}\left(\frac{1}{z+}\frac{p}{1+}\frac{1}{z+}\frac{p+1}{1+}\frac{2}{z+}\mathrm{\cdots }\right),$$ .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{ph}$: phase, $z$: complex variable and $p$: parameter

A&S Ref:

5.1.22

Permalink:

http://dlmf.nist.gov/8.19.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(vii), §8.19 and Ch.8

… ►

8.19.23 $${\int }_z^{\mathrm{\infty }}{E}_{p-1}\left(t\right)dt=E_p\left(z\right),$$ ,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable and $p$: parameter

Referenced by:

§8.19(x)

Permalink:

http://dlmf.nist.gov/8.19.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(x), §8.19 and Ch.8

…

10: 4.2 Definitions

… ►

4.2.3 $$\ln z=\ln \left|z\right|+\mathrm{i}\mathrm{ph}z,$$ .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

4.1.2

Referenced by:

§4.2(i)

Permalink:

http://dlmf.nist.gov/4.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(i), §4.2 and Ch.4

… ►

4.2.6 $$\mathrm{Ln}z=\ln z+2k\pi \mathrm{i},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: integer and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/4.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(i), §4.2 and Ch.4

►where $k$ is the excess of the number of times the path in (4.2.1) crosses the negative real axis in the positive sense over the number of times in the negative sense. … ►

4.2.20 $$\exp \left(z+2\pi \mathrm{i}\right)=\exp z.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.2.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(iii), §4.2 and Ch.4

… ►

4.2.23 $$\mathrm{ph}\left(\exp z\right)=\mathrm{\Im }z+2k\pi ,$$ $k\in \mathbb{Z}$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\exp z$: exponential function, $\mathrm{\Im }$: imaginary part, $\mathbb{Z}$: set of all integers, $\mathrm{ph}$: phase, $k$: integer and $z$: complex variable

A&S Ref:

4.2.14 (modified)

Permalink:

http://dlmf.nist.gov/4.2.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(iii), §4.2 and Ch.4

…