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11: 7.4 Symmetry

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$C\left(\mathrm{i}z\right)=\mathrm{i}C\left(z\right),$

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$S\left(\mathrm{i}z\right)=-\mathrm{i}S\left(z\right).$

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$\mathrm{f}\left(\mathrm{i}z\right)=(1/\sqrt{2}){\mathrm{e}}^{\frac{1}{4}\pi \mathrm{i}-\frac{1}{2}\pi \mathrm{i}{z}^2}-\mathrm{i}\mathrm{f}\left(z\right),$

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$\mathrm{g}\left(\mathrm{i}z\right)=(1/\sqrt{2}){\mathrm{e}}^{-\frac{1}{4}\pi \mathrm{i}-\frac{1}{2}\pi \mathrm{i}{z}^2}+\mathrm{i}\mathrm{g}\left(z\right).$

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12: 9.3 Graphics

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Figure 9.3.3: $\mathrm{Ai}\left(x+\mathrm{i}y\right)$. Magnify 3D Help

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Figure 9.3.4: $\mathrm{Bi}\left(x+\mathrm{i}y\right)$. Magnify 3D Help

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Figure 9.3.5: ${\mathrm{Ai}}^{\prime }\left(x+\mathrm{i}y\right)$. Magnify 3D Help

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Figure 9.3.6: ${\mathrm{Bi}}^{\prime }\left(x+\mathrm{i}y\right)$. Magnify 3D Help

13: 14.22 Graphics

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Figure 14.22.1: $P_{1/2}^0\left(x+\mathrm{i}y\right)$, $-5\le x\le 5$, $-5\le y\le 5$. … Magnify 3D Help

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Figure 14.22.2: $P_{1/2}^{-1/2}\left(x+\mathrm{i}y\right)$, $-5\le x\le 5$, $-5\le y\le 5$. … Magnify 3D Help

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Figure 14.22.3: $P_{1/2}^{-1}\left(x+\mathrm{i}y\right)$, $-5\le x\le 5$, $-5\le y\le 5$. … Magnify 3D Help

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Figure 14.22.4: ${𝑸}_0^0\left(x+\mathrm{i}y\right)$, $-5\le x\le 5$, $-5\le y\le 5$. … Magnify 3D Help

14: 29.10 Lamé Functions with Imaginary Periods

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29.10.2 $$z^{\prime }=\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),$$

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Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.10 and Ch.29

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${\mathrm{𝐸𝑐}}_{\nu }^{2m}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k_{}^{\prime }{}_{}{}^2),$

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${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k_{}^{\prime }{}_{}{}^2),$

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${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k_{}^{\prime }{}_{}{}^2),$

… ►The first and the fourth functions have period $2\mathrm{i}{K}^{\prime }$; the second and the third have period $4\mathrm{i}{K}^{\prime }$. …

15: 32 Painlevé Transcendents

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16: 10.34 Analytic Continuation

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10.34.1 $$I_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{m\nu \pi \mathrm{i}}{I}_{\nu }\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $m$: integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.30

Referenced by:

§10.30(ii), §10.34, §10.47(v)

Permalink:

http://dlmf.nist.gov/10.34.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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10.34.2 $$K_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{-m\nu \pi \mathrm{i}}{K}_{\nu }\left(z\right)-\pi \mathrm{i}\sin \left(m\nu \pi \right)\csc \left(\nu \pi \right)I_{\nu }\left(z\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.31

Referenced by:

§10.27, §10.34, §10.34

Permalink:

http://dlmf.nist.gov/10.34.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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10.34.3 $$I_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=(\mathrm{i}/\pi )\left(\pm {\mathrm{e}}^{m\nu \pi \mathrm{i}}{K}_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp {\mathrm{e}}^{(m\mp 1)\nu \pi \mathrm{i}}{K}_{\nu }\left(z\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $m$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.34, §10.40(i), §10.40(iii), §10.41(iv)

Permalink:

http://dlmf.nist.gov/10.34.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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10.34.4 $$K_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=\csc \left(\nu \pi \right)\left(\pm \sin \left(m\nu \pi \right)K_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp \sin \left((m\mp 1)\nu \pi \right)K_{\nu }\left(z\right)\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.34, §10.40(i)

Permalink:

http://dlmf.nist.gov/10.34.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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10.34.5 $$K_n\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn}{K}_n\left(z\right)+{(-1)}^{n(m-1)-1}m\pi \mathrm{i}{I}_n\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $m$: integer, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.34.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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17: 29.14 Orthogonality

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29.14.3 $$w(s,t)={\mathrm{sn}}^2(K+\mathrm{i}t,k)-{\mathrm{sn}}^2(s,k).$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $k$: real parameter and $w(s,t)$: function

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

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29.14.4 $${\mathrm{𝑠𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑠𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

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Symbols:

${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

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29.14.5 $${\mathrm{𝑐𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑐𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

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Symbols:

${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

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29.14.6 $${\mathrm{𝑑𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑑𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

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Symbols:

${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

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29.14.7 $${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(s,k^2){\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(K+\mathrm{i}t,k^2),$$

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Symbols:

${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

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18: 4.4 Special Values and Limits

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4.4.2 $$\ln \left(-1\pm \mathrm{i}0\right)=\pm \pi \mathrm{i},$$

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Symbols:

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

A&S Ref:

4.1.14 (modified)

Permalink:

http://dlmf.nist.gov/4.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(i), §4.4 and Ch.4

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4.4.3 $$\ln \left(\pm \mathrm{i}\right)=\pm \frac{1}{2}\pi \mathrm{i}.$$

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Symbols:

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

A&S Ref:

4.1.15

Permalink:

http://dlmf.nist.gov/4.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(i), §4.4 and Ch.4

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4.4.5 $${\mathrm{e}}^{\pm \pi \mathrm{i}}=-1,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

A&S Ref:

4.2.26

Permalink:

http://dlmf.nist.gov/4.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(ii), §4.4 and Ch.4

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4.4.6 $${\mathrm{e}}^{\pm \pi \mathrm{i}/2}=\pm \mathrm{i},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

A&S Ref:

4.2.27

Permalink:

http://dlmf.nist.gov/4.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(ii), §4.4 and Ch.4

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4.4.8 $${\mathrm{e}}^{\pm \pi \mathrm{i}/3}=\frac{1}{2}\pm \mathrm{i}\frac{\sqrt{3}}{2},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

Permalink:

http://dlmf.nist.gov/4.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(ii), §4.4 and Ch.4

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19: 7.5 Interrelations

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7.5.2 $$C\left(z\right)+\mathrm{i}S\left(z\right)=\frac{1}{2}(1+\mathrm{i})-ℱ\left(z\right).$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $ℱ\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.5 $${\mathrm{e}}^{-\frac{1}{2}\pi \mathrm{i}{z}^2}ℱ\left(z\right)=\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right).$$

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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, $ℱ\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.13(iv), §7.5, §7.7(ii)

Permalink:

http://dlmf.nist.gov/7.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.7 $$\zeta =\frac{1}{2}\sqrt{\pi }(1\mp \mathrm{i})z,$$

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Defines:

$\zeta$: change of variable (locally)

Symbols:

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

Referenced by:

§7.12(ii), §7.5

Permalink:

http://dlmf.nist.gov/7.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.8 $$C\left(z\right)\pm \mathrm{i}S\left(z\right)=\frac{1}{2}(1\pm \mathrm{i})\mathrm{erf}\zeta .$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\mathrm{erf}z$: error function, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

Referenced by:

§7.5, §7.5, §7.6(i)

Permalink:

http://dlmf.nist.gov/7.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.10 $$\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right)=\frac{1}{2}(1\pm \mathrm{i}){\mathrm{e}}^{{\zeta }^2}\mathrm{erfc}\zeta .$$

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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, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

A&S Ref:

7.3.22 (in different form)

Referenced by:

§7.12(ii), §7.13(iv), §7.5, §7.5

Permalink:

http://dlmf.nist.gov/7.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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20: 33.7 Integral Representations

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33.7.1 $$F_{\mathrm{\ell }}(\eta ,\rho )=\frac{{\rho }^{\mathrm{\ell }+1}{2}^{\mathrm{\ell }}{\mathrm{e}}^{\mathrm{i}\rho -(\pi \eta /2)}}{|\mathrm{\Gamma }\left(\mathrm{\ell }+1+\mathrm{i}\eta \right)|}{\int }_0^1{\mathrm{e}}^{-2\mathrm{i}\rho t}{t}^{\mathrm{\ell }+\mathrm{i}\eta }{(1-t)}^{\mathrm{\ell }-\mathrm{i}\eta }dt,$$

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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, $\mathrm{i}$: imaginary unit, $\int$: integral, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Referenced by:

§33.7, §33.7

Permalink:

http://dlmf.nist.gov/33.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.7 and Ch.33

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33.7.2 $$H_{\mathrm{\ell }}^{-}(\eta ,\rho )=\frac{{\mathrm{e}}^{-\mathrm{i}\rho }{\rho }^{-\mathrm{\ell }}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{\mathrm{\ell }-\mathrm{i}\eta }{(t+2\mathrm{i}\rho )}^{\mathrm{\ell }+\mathrm{i}\eta }dt,$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

A&S Ref:

14.3.1 (modified)

Referenced by:

§33.7

Permalink:

http://dlmf.nist.gov/33.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.7 and Ch.33

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33.7.3 $$H_{\mathrm{\ell }}^{-}(\eta ,\rho )=\frac{-\mathrm{i}{\mathrm{e}}^{-\pi \eta }{\rho }^{\mathrm{\ell }+1}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{\int }_0^{\mathrm{\infty }}\left(\frac{\exp \left(-\mathrm{i}(\rho \tanh t-2\eta t)\right)}{{(\cosh t)}^{2\mathrm{\ell }+2}}+\mathrm{i}{(1+t^2)}^{\mathrm{\ell }}\exp \left(-\rho t+2\eta \arctan t\right)\right)dt,$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\arctan z$: arctangent function, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

A&S Ref:

14.3.3

Referenced by:

§33.7

Permalink:

http://dlmf.nist.gov/33.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.7 and Ch.33

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33.7.4 $$H_{\mathrm{\ell }}^{+}(\eta ,\rho )=\frac{\mathrm{i}{\mathrm{e}}^{-\pi \eta }{\rho }^{\mathrm{\ell }+1}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{\int }_{-1}^{-\mathrm{i}\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{i}\rho t}{(1-t)}^{\mathrm{\ell }-\mathrm{i}\eta }{(1+t)}^{\mathrm{\ell }+\mathrm{i}\eta }dt.$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

A&S Ref:

14.3.2

Referenced by:

§33.7, §33.7

Permalink:

http://dlmf.nist.gov/33.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.7 and Ch.33

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