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1: 7.5 Interrelations

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7.5.6 $${\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}{z}^2}(\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right))=\frac{1}{2}(1\pm \mathrm{i})-(C\left(z\right)\pm \mathrm{i}S\left(z\right)).$$

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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, $C\left(z\right)$: Fresnel integral, $S\left(z\right)$: 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:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E6

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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7.5.9 $$C\left(z\right)\pm \mathrm{i}S\left(z\right)=\frac{1}{2}(1\pm \mathrm{i})\left(1-{\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}{z}^2}w\left(\mathrm{i}\zeta \right)\right).$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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2: 32.9 Other Elementary Solutions

… ► ${\text{P}}_{\text{V}}$, with $\alpha =\beta =0$ and ${\gamma }^2+2\delta =0$, has solutions $w(z)=C\exp \left(\pm \sqrt{-2\delta }z\right)$, with $C$ an arbitrary constant. …

3: 10.6 Recurrence Relations and Derivatives

… ►For results on modified quotients of the form $z{𝒞}_{\nu \pm 1}\left(z\right)/{𝒞}_{\nu }\left(z\right)$ see Onoe (1955) and Onoe (1956). …

4: 10.23 Sums

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10.23.1 $${𝒞}_{\nu }\left(\lambda z\right)={\lambda }^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}\frac{{(\mp 1)}^k{({\lambda }^2-1)}^k{(\frac{1}{2}z)}^k}{k!}{𝒞}_{\nu \pm k}\left(z\right),$$ .

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $!$: factorial (as in $n!$), $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.74

Referenced by:

§10.44(i), §10.66

Permalink:

http://dlmf.nist.gov/10.23.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(i), §10.23 and Ch.10

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10.23.2 $${𝒞}_{\nu }\left(u\pm v\right)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{𝒞}_{\nu \mp k}\left(u\right)J_k\left(v\right),$$ .

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${𝒞}_{\nu }\left(z\right)$: cylinder function, $k$: nonnegative integer and $\nu$: complex parameter

A&S Ref:

9.1.75

Referenced by:

§10.23(iii), §10.23(ii), §10.44(ii), §10.66

Permalink:

http://dlmf.nist.gov/10.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

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5: 10.13 Other Differential Equations

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10.13.4 $$w^{\prime \prime }+\frac{1\mp 2\nu }{z}{w}^{\prime }+{\lambda }^{2}w=0,$$ $w=z^{\pm \nu }{𝒞}_{\nu }\left(\lambda z\right)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.52

Referenced by:

Erratum (V1.0.8) for Equation (10.13.4)

Permalink:

http://dlmf.nist.gov/10.13.E4

Encodings:

pMML, png, TeX

Generalization (effective with 1.0.8):

This equation has been generalized to allow an additional case. Originally only the case $w=z^{\nu }{𝒞}_{\nu }\left(\lambda z\right)$ was covered.

Suggested 2014-01-27 by Tom Koornwinder

See also:

Annotations for §10.13 and Ch.10

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6: 11.10 Anger–Weber Functions

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$A_{\pm }(\chi )=C\left(\chi \right)\pm S\left(\chi \right),$

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7: 10.22 Integrals

… ►In this subsection ${𝒞}_{\nu }\left(z\right)$ and ${𝒟}_{\mu }(z)$ denote cylinder functions(§10.2(ii)) of orders $\nu$ and $\mu$, respectively, not necessarily distinct. ►

${\displaystyle \int z^{\nu +1}{𝒞}_{\nu }\left(z\right)dz}=z^{\nu +1}{𝒞}_{\nu +1}\left(z\right),$

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${\displaystyle \int z^{-\nu +1}{𝒞}_{\nu }\left(z\right)dz}=-z^{-\nu +1}{𝒞}_{\nu -1}\left(z\right).$

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$$\int {\mathrm{e}}^{\mathrm{i}z}{z}^{\nu }{𝒞}_{\nu }\left(z\right)dz=\frac{{\mathrm{e}}^{\mathrm{i}z}{z}^{\nu +1}}{2\nu +1}({𝒞}_{\nu }\left(z\right)-\mathrm{i}{𝒞}_{\nu +1}\left(z\right)),$$ $\nu \ne -\frac{1}{2}$,

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10.22.5 $$\int z{𝒞}_{\mu }\left(az\right){𝒟}_{\mu }(az)dz=\frac{1}{4}{z}^2\left(2{𝒞}_{\mu }\left(az\right){𝒟}_{\mu }(az)-{𝒞}_{\mu -1}\left(az\right){𝒟}_{\mu +1}(az)-{𝒞}_{\mu +1}\left(az\right){𝒟}_{\mu -1}(az)\right),$$

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable and ${𝒟}_{\nu }(z)$: cylinder function

Referenced by:

§10.22(ii)

Permalink:

http://dlmf.nist.gov/10.22.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(i), §10.22(i), §10.22 and Ch.10

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8: 19.3 Graphics

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Figure 19.3.2: $R_C(x,1)$ and the Cauchy principal value of $R_C(x,-1)$ for $0\le x\le 5$. …Note that $R_C(x,\pm y)=y^{-1/2}{R}_C(x/y,\pm 1)$, $y>0$. Magnify

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9: 18.10 Integral Representations

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Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).

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$p_n(x)$	$g_0(x)$	$g_1(z,x)$	$g_2(z,x)$	$c$	Conditions
$P_n^{(\alpha ,\beta )}\left(x\right)$	${(1-x)}^{-\alpha }{(1+x)}^{-\beta }$	${\displaystyle \frac{z^2-1}{2(z-x)}}$	${(1-z)}^{\alpha }{(1+z)}^{\beta }$	$x$	$\pm 1$ outside $C$.
$C_n^{(\lambda )}\left(x\right)$	$1$	$z^{-1}$	${(1-2xz+z^2)}^{-\lambda }$	$0$	${\mathrm{e}}^{\pm \mathrm{i}\theta }$ outside $C$ (where $x=\cos \theta$).
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10: 32.10 Special Function Solutions

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32.10.19 $$\varphi (z)=\left(C_{1}U(a,\sqrt{2}z)+C_{2}V(a,\sqrt{2}z)\right)\exp \left(\frac{1}{2}\epsilon z^2\right),$$

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Symbols:

$\exp z$: exponential function, $U(a,z)$: parabolic cylinder function, $V(a,z)$: parabolic cylinder function, $z$: real, $\epsilon =\pm 1$ , $\varphi (z)$: function, $a$ and $C$: constant

Permalink:

http://dlmf.nist.gov/32.10.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §32.10(iv), §32.10 and Ch.32

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32.10.22

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $z$: real, $\epsilon =\pm 1$ and $C$: constant

Permalink:

http://dlmf.nist.gov/32.10.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §32.10(iv), §32.10 and Ch.32

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