normalizing constant

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1: 33.13 Complex Variable and Parameters

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33.13.1 $$C_{\mathrm{\ell }}\left(\eta \right)=2^{\mathrm{\ell }}{\mathrm{e}}^{\mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)-(\pi \eta /2)}\mathrm{\Gamma }\left(\mathrm{\ell }+1-\mathrm{i}\eta \right)/\mathrm{\Gamma }\left(2\mathrm{\ell }+2\right),$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

Permalink:

http://dlmf.nist.gov/33.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.13 and Ch.33

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33.13.2 $$R_{\mathrm{\ell }}=(2\mathrm{\ell }+1)C_{\mathrm{\ell }}\left(\eta \right)/C_{\mathrm{\ell }-1}\left(\eta \right).$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\eta$: real parameter and $R_{\mathrm{\ell }}$: factor

Permalink:

http://dlmf.nist.gov/33.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.13 and Ch.33

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2: 30.1 Special Notation

… ►where $d_{mn}(\gamma )$ is a normalization constant determined by …

3: 31.9 Orthogonality

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31.9.2 $${\int }_{\zeta }^{(1+,0+,1-,0-)}{t}^{\gamma -1}{(1-t)}^{\delta -1}{(t-a)}^{ϵ-1}{w}_m(t)w_k(t)dt={\delta }_{m,k}{\theta }_m.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $\zeta$: point and ${\theta }_m$: normalization constant

Permalink:

http://dlmf.nist.gov/31.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.9(i), §31.9 and Ch.31

… ►The normalization constant ${\theta }_m$ is given by ►

31.9.3 $${\theta }_m={(1-{\mathrm{e}}^{2\pi \mathrm{i}\gamma })(1-{\mathrm{e}}^{2\pi \mathrm{i}\delta }){\zeta }^{\gamma }{(1-\zeta )}^{\delta }{(\zeta -a)}^{ϵ}\frac{f_0(q,\zeta )}{f_1(q,\zeta )}\frac{\partial }{\partial q}𝒲\left\{f_0(q,\zeta ),f_1(q,\zeta )\right\}|}_{q=q_m},$$

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Defines:

${\theta }_m$: normalization constant (locally)

Symbols:

$𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\zeta$: point and $f_0(q_m,z)$, $f_1(q_m,z)$: coefficients

Permalink:

http://dlmf.nist.gov/31.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.9(i), §31.9 and Ch.31

… ►For further information, including normalization constants, see Sleeman (1966a). …

4: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

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33.5.6 $$C_{\mathrm{\ell }}\left(0\right)=\frac{2^{\mathrm{\ell }}\mathrm{\ell }!}{(2\mathrm{\ell }+1)!}=\frac{1}{(2\mathrm{\ell }+1)!!}.$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!!$: double factorial, $!$: factorial (as in $n!$) and $\mathrm{\ell }$: nonnegative integer

Permalink:

http://dlmf.nist.gov/33.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.5(ii), §33.5 and Ch.33

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33.5.9 $$C_{\mathrm{\ell }}\left(\eta \right)\sim \frac{{\mathrm{e}}^{-\pi \eta /2}}{(2\mathrm{\ell }+1)!!}\sim {\mathrm{e}}^{-\pi \eta /2}\frac{{\mathrm{e}}^{\mathrm{\ell }}}{\sqrt{2}{(2\mathrm{\ell })}^{\mathrm{\ell }+1}}.$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!!$: double factorial, $\mathrm{e}$: base of natural logarithm, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

Referenced by:

§33.23(iv), §33.5(iv)

Permalink:

http://dlmf.nist.gov/33.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §33.5(iv), §33.5 and Ch.33

5: 33.7 Integral Representations

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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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6: 33.2 Definitions and Basic Properties

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33.2.3 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)2^{-\mathrm{\ell }-1}{(\mp \mathrm{i})}^{\mathrm{\ell }+1}{M}_{\pm \mathrm{i}\eta ,\mathrm{\ell }+\frac{1}{2}}\left(\pm 2\mathrm{i}\rho \right),$$

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Defines:

$F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Referenced by:

§13.18(vi), §33.2(ii)

Permalink:

http://dlmf.nist.gov/33.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.2(ii), §33.2 and Ch.33

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33.2.5 $$C_{\mathrm{\ell }}\left(\eta \right)=\frac{2^{\mathrm{\ell }}{\mathrm{e}}^{-\pi \eta /2}|\mathrm{\Gamma }\left(\mathrm{\ell }+1+\mathrm{i}\eta \right)|}{(2\mathrm{\ell }+1)!}.$$

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Defines:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

A&S Ref:

14.1.7

Referenced by:

§33.10(ii), §33.10(iii), §33.16(i)

Permalink:

http://dlmf.nist.gov/33.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §33.2(ii), §33.2 and Ch.33

… ► $F_{\mathrm{\ell }}(\eta ,\rho )$ is a real and analytic function of $\rho$ on the open interval , and also an analytic function of $\eta$ when . ►The normalizing constant $C_{\mathrm{\ell }}\left(\eta \right)$ is always positive, and has the alternative form ►

33.2.6 $$C_{\mathrm{\ell }}\left(\eta \right)=\frac{2^{\mathrm{\ell }}{\left((2\pi \eta /({\mathrm{e}}^{2\pi \eta }-1)){\prod }_{k=1}^{\mathrm{\ell }}({\eta }^2+k^2)\right)}^{1/2}}{(2\mathrm{\ell }+1)!}.$$

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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, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

Referenced by:

§33.13, §33.16(i)

Permalink:

http://dlmf.nist.gov/33.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.2(ii), §33.2 and Ch.33

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7: 33.9 Expansions in Series of Bessel Functions

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33.9.3 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\frac{(2\mathrm{\ell }+1)!}{{(2\eta )}^{2\mathrm{\ell }+1}}{\rho }^{-\mathrm{\ell }}\sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{b}_k{t}^{k/2}{I}_k\left(2\sqrt{t}\right),$$ $\eta >0$,

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $t$: parameter and $b_k$: coefficient

A&S Ref:

14.4.1

Referenced by:

§33.9(ii), §33.9(ii)

Permalink:

http://dlmf.nist.gov/33.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.9(ii), §33.9 and Ch.33

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33.9.4 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\frac{(2\mathrm{\ell }+1)!}{{(2\left|\eta \right|)}^{2\mathrm{\ell }+1}}{\rho }^{-\mathrm{\ell }}\sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{b}_k{t}^{k/2}{J}_k\left(2\sqrt{t}\right),$$ .

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!$: factorial (as in $n!$), $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $t$: parameter and $b_k$: coefficient

Referenced by:

§33.9(ii), §33.9(ii)

Permalink:

http://dlmf.nist.gov/33.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.9(ii), §33.9 and Ch.33

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33.9.6 $$G_{\mathrm{\ell }}(\eta ,\rho )\sim \frac{{\rho }^{-\mathrm{\ell }}}{(\mathrm{\ell }+\frac{1}{2}){\lambda }_{\mathrm{\ell }}(\eta )C_{\mathrm{\ell }}\left(\eta \right)}\sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{(-1)}^k{b}_k{t}^{k/2}{K}_k\left(2\sqrt{t}\right),$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\sim$: asymptotic equality, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $t$: parameter, $b_k$: coefficient and ${\lambda }_{\mathrm{\ell }}(\eta )$: coefficient

A&S Ref:

14.4.2

Referenced by:

§33.9(ii)

Permalink:

http://dlmf.nist.gov/33.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.9(ii), §33.9 and Ch.33

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8: 33.6 Power-Series Expansions in $\rho$

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33.6.1 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\sum _{k=\mathrm{\ell }+1}^{\mathrm{\infty }}{A}_k^{\mathrm{\ell }}(\eta ){\rho }^k,$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient

A&S Ref:

14.1.4, 14.1.5

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.6 and Ch.33

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33.6.2 $$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\sum _{k=\mathrm{\ell }+1}^{\mathrm{\infty }}k{A}_k^{\mathrm{\ell }}(\eta ){\rho }^{k-1},$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient

A&S Ref:

14.1.4, 14.1.5

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.6 and Ch.33

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9: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

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10: 33.16 Connection Formulas

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33.16.1 $$F_{\mathrm{\ell }}(\eta ,\rho )=\frac{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{{(-2\eta )}^{\mathrm{\ell }+1}}f(1/{\eta }^2,\mathrm{\ell };-\eta \rho ),$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $!$: factorial (as in $n!$), $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Referenced by:

§33.18

Permalink:

http://dlmf.nist.gov/33.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.16(i), §33.16 and Ch.33

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33.16.2 $$G_{\mathrm{\ell }}(\eta ,\rho )=\frac{\pi {(-2\eta )}^{\mathrm{\ell }}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}h(1/{\eta }^2,\mathrm{\ell };-\eta \rho ),$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Referenced by:

§33.18

Permalink:

http://dlmf.nist.gov/33.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.16(i), §33.16 and Ch.33

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