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1: 10.24 Functions of Imaginary Order

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$${\tilde{J}}_{\nu }\left(x\right)=\mathrm{sech}\left(\frac{1}{2}\pi \nu \right)\mathrm{\Re }\left(J_{\mathrm{i}\nu }\left(x\right)\right),$$

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$${\tilde{Y}}_{\nu }\left(x\right)=\mathrm{sech}\left(\frac{1}{2}\pi \nu \right)\mathrm{\Re }\left(Y_{\mathrm{i}\nu }\left(x\right)\right),$$

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10.24.5 $$𝒲\left\{{\tilde{J}}_{\nu }\left(x\right),{\tilde{Y}}_{\nu }\left(x\right)\right\}=2/(\pi x).$$

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Symbols:

${\tilde{J}}_{\nu }\left(x\right)$: Bessel function of imaginary order, ${\tilde{Y}}_{\nu }\left(x\right)$: Bessel function of imaginary order, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $x$: real variable and $\nu$: complex parameter

Referenced by:

§10.24

Permalink:

http://dlmf.nist.gov/10.24.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.24 and Ch.10

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$${\tilde{Y}}_{\nu }\left(x\right)=\sqrt{2/(\pi x)}\sin \left(x-\frac{1}{4}\pi \right)+O\left(x^{-\frac{3}{2}}\right).$$

… ►Also, in consequence of (10.24.7)–(10.24.9), when $x$ is small either ${\tilde{J}}_{\nu }\left(x\right)$ and $\tanh \left(\frac{1}{2}\pi \nu \right){\tilde{Y}}_{\nu }\left(x\right)$ or ${\tilde{J}}_{\nu }\left(x\right)$ and ${\tilde{Y}}_{\nu }\left(x\right)$ comprise a numerically satisfactory pair depending whether $\nu \ne 0$ or $\nu =0$. …

2: 10.45 Functions of Imaginary Order

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10.45.4 $$𝒲\left\{{\tilde{K}}_{\nu }\left(x\right),{\tilde{I}}_{\nu }\left(x\right)\right\}=1/x.$$

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Symbols:

$𝒲$: Wronskian, ${\tilde{I}}_{\nu }\left(x\right)$: modified Bessel function of the first kind of imaginary order, ${\tilde{K}}_{\nu }\left(x\right)$: modified Bessel function fo the second kind of imaginary order, $x$: real variable and $\nu$: complex parameter

Referenced by:

§10.45

Permalink:

http://dlmf.nist.gov/10.45.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.45 and Ch.10

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$${\tilde{I}}_{\nu }\left(x\right)={(2\pi x)}^{-\frac{1}{2}}{\mathrm{e}}^x\left(1+O\left(x^{-1}\right)\right),$$

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$${\tilde{K}}_{\nu }\left(x\right)={(\pi /(2x))}^{\frac{1}{2}}{\mathrm{e}}^{-x}\left(1+O\left(x^{-1}\right)\right).$$

… ►In consequence of (10.45.5)–(10.45.7), ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.45.1) when $x$ is large, and either ${\tilde{I}}_{\nu }\left(x\right)$ and $(1/\pi )\sinh \left(\pi \nu \right){\tilde{K}}_{\nu }\left(x\right)$, or ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$, comprise a numerically satisfactory pair when $x$ is small, depending whether $\nu \ne 0$ or $\nu =0$. … ►In this reference ${\tilde{I}}_{\nu }\left(x\right)$ is denoted by $(1/\pi )\sinh \left(\pi \nu \right)L_{\mathrm{i}\nu }(x)$. …

3: 4.15 Graphics

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	A	B	C	$\overline{\mathrm{C}}$	D	$\overline{\mathrm{D}}$	E	$\overline{\mathrm{E}}$	F
$z$	$0$	$\frac{1}{2}\pi$	$\frac{1}{2}\pi +\mathrm{i}r$	$\frac{1}{2}\pi -\mathrm{i}r$	$\mathrm{i}r$	$-\mathrm{i}r$	$-\frac{1}{2}\pi +\mathrm{i}r$	$-\frac{1}{2}\pi -\mathrm{i}r$	$-\frac{1}{2}\pi$
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Figure 4.15.7: Conformal mapping of sine and inverse sine. … Magnify

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4.15.1 $$\cos \left(x+\mathrm{i}y\right)=\sin \left(x+\frac{1}{2}\pi +\mathrm{i}y\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.15(iii), §4.15 and Ch.4

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4.15.2 $$\cot \left(x+\mathrm{i}y\right)=-\tan \left(x+\frac{1}{2}\pi +\mathrm{i}y\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\mathrm{i}$: imaginary unit, $\tan z$: tangent function, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.15(iii), §4.15 and Ch.4

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4.15.3 $$\sec \left(x+\mathrm{i}y\right)=\csc \left(x+\frac{1}{2}\pi +\mathrm{i}y\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{i}$: imaginary unit, $\sec z$: secant function, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.15(iii), §4.15 and Ch.4

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4: 4.31 Special Values and Limits

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Table 4.31.1: Hyperbolic functions: values at multiples of $\frac{1}{2}\pi \mathrm{i}$.

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$z$	0	$\frac{1}{2}\pi \mathrm{i}$	$\pi \mathrm{i}$	$\frac{3}{2}\pi \mathrm{i}$	$\mathrm{\infty }$
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5: 6.5 Further Interrelations

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6.5.2 $$\mathrm{Ei}\left(x\right)=-\frac{1}{2}(E_1\left(-x+\mathrm{i}0\right)+E_1\left(-x-\mathrm{i}0\right)),$$

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Symbols:

$E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $\mathrm{i}$: imaginary unit and $x$: real variable

A&S Ref:

5.1.7

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

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6.5.3 $$\frac{1}{2}(\mathrm{Ei}\left(x\right)+E_1\left(x\right))=\mathrm{Shi}\left(x\right)=-\mathrm{i}\mathrm{Si}\left(\mathrm{i}x\right),$$

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Symbols:

$E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $\mathrm{Shi}\left(z\right)$: hyperbolic sine integral, $\mathrm{i}$: imaginary unit, $\mathrm{Si}\left(z\right)$: sine integral and $x$: real variable

A&S Ref:

5.2.22 (in modified form)

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

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6.5.4 $$\frac{1}{2}(\mathrm{Ei}\left(x\right)-E_1\left(x\right))=\mathrm{Chi}\left(x\right)=\mathrm{Ci}\left(\mathrm{i}x\right)-\frac{1}{2}\pi \mathrm{i}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $\mathrm{Chi}\left(z\right)$: hyperbolic cosine integral, $\mathrm{i}$: imaginary unit and $x$: real variable

A&S Ref:

5.2.24 (in modified form)

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

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6.5.5 $$\mathrm{Si}\left(z\right)=\frac{1}{2}\mathrm{i}(E_1\left(-\mathrm{i}z\right)-E_1\left(\mathrm{i}z\right))+\frac{1}{2}\pi ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\mathrm{Si}\left(z\right)$: sine integral and $z$: complex variable

A&S Ref:

5.2.21

Referenced by:

§6.12(ii), §6.5

Permalink:

http://dlmf.nist.gov/6.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

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6.5.6 $$\mathrm{Ci}\left(z\right)=-\frac{1}{2}(E_1\left(\mathrm{i}z\right)+E_1\left(-\mathrm{i}z\right)),$$

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Symbols:

$\mathrm{Ci}\left(z\right)$: cosine integral, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

5.2.23

Referenced by:

§6.12(ii), §6.5

Permalink:

http://dlmf.nist.gov/6.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

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6: 10.26 Graphics

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Figure 10.26.7: ${\tilde{I}}_{1/2}\left(x\right)$, ${\tilde{K}}_{1/2}\left(x\right)$, $0.01\le x\le 3$. Magnify

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7: 4.28 Definitions and Periodicity

… ►The zeros of $\sinh z$ and $\cosh z$ are $z=\mathrm{i}k\pi$ and $z=\mathrm{i}\left(k+\frac{1}{2}\right)\pi$, respectively, $k\in \mathbb{Z}$.

8: 7.4 Symmetry

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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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9: 29.10 Lamé Functions with Imaginary Periods

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

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

10: 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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