DLMF:33.7.E3 (Q9540): Difference between revisions

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Revision as of 14:25, 2 January 2020 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Changed an Item: Add constraint ← Older edit		Latest revision as of 14:25, 2 January 2020 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Changed an Item: Add constraint
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		Property / Symbols used
			hyperbolic tangent function
		Property / Symbols used: hyperbolic tangent function / rank
			Normal rank
		Property / Symbols used: hyperbolic tangent function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\tanh\NVar{z}}}$ \tanh@@{\NVar{z}}
		Property / Symbols used: hyperbolic tangent function / qualifier
			xml-id: C4.S28.E4.m2adec
		Property / Symbols used
			imaginary unit
		Property / Symbols used: imaginary unit / rank
			Normal rank
		Property / Symbols used: imaginary unit / qualifier
			Defining formula: ${\displaystyle{\displaystyle\mathrm{i}}}$ \iunit
		Property / Symbols used: imaginary unit / qualifier
			xml-id: C1.S9.E1.m2abdec
		Property / Symbols used
			Q10771
		Property / Symbols used: Q10771 / rank
			Normal rank
		Property / Symbols used: Q10771 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\int}}$ \int
		Property / Symbols used: Q10771 / qualifier
			xml-id: C1.S4.SS4.m3abdec
		Property / Symbols used
			Q10816
		Property / Symbols used: Q10816 / rank
			Normal rank
		Property / Symbols used: Q10816 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\operatorname{arctan}\NVar{z}}}$ \atan@@{\NVar{z}}
		Property / Symbols used: Q10816 / qualifier
			xml-id: C4.S23.SS2.p1.m7adec
		Property / Symbols used
			irregular Coulomb radial functions
		Property / Symbols used: irregular Coulomb radial functions / rank
			Normal rank
		Property / Symbols used: irregular Coulomb radial functions / qualifier
			Defining formula: ${\displaystyle{\displaystyle{H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)}}$ \irregCoulombH{\NVar{\pm}}{\NVar{\ell}}@{\NVar{\eta}}{\NVar{\rho}}
		Property / Symbols used: irregular Coulomb radial functions / qualifier
			xml-id: C33.S2.E7.m2aadec
		Property / Symbols used
			Q12666
		Property / Symbols used: Q12666 / rank
			Normal rank
		Property / Symbols used: Q12666 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\ell}}$ \ell
		Property / Symbols used: Q12666 / qualifier
			xml-id: C33.S1.XMD2.m1bdec
		Property / Symbols used
			Q12667
		Property / Symbols used: Q12667 / rank
			Normal rank
		Property / Symbols used: Q12667 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\rho}}$ \rho
		Property / Symbols used: Q12667 / qualifier
			xml-id: C33.S1.XMD5.m1bdec
		Property / Symbols used
			Q12668
		Property / Symbols used: Q12668 / rank
			Normal rank
		Property / Symbols used: Q12668 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\eta}}$ \eta
		Property / Symbols used: Q12668 / qualifier
			xml-id: C33.S1.XMD7.m1bdec

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Latest revision as of 14:25, 2 January 2020

No description defined

Language	Label	Description	Also known as
English	DLMF:33.7.E3	No description defined

Statements

Defining formula

${\displaystyle{\displaystyle{H^{-}_{\ell}}\left(\eta,\rho\right)=\frac{-\mathrm{i}e^{-\pi\eta}\rho^{\ell+1}}{(2\ell+1)!C_{\ell}\left(\eta\right)}\int_{0}^{\infty}\left(\frac{\exp\left(-\mathrm{i}(\rho\tanh t-2\eta t)\right)}{(\cosh t)^{2\ell+2}}+\mathrm{i}(1+t^{2})^{\ell}\exp\left(-\rho t+2\eta\operatorname{arctan}t\right)\right)\mathrm{d}t,}}$

0 references

DLMF id

DLMF:33.7.E3

0 references

Symbols used

normalizing constant for Coulomb radial functions

Defining formula

${\displaystyle{\displaystyle C_{\NVar{\ell}}\left(\NVar{\eta}\right)}}$

xml-id

C33.S2.E5.m2aadec

0 references

the ratio of the circumference of a circle to its diameter

Defining formula

${\displaystyle{\displaystyle\pi}}$

xml-id

C3.S12.E1.m2aadec

0 references

Q10770

Defining formula

${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$

xml-id

C1.S4.SS4.m1abdec

0 references

exponential function

Defining formula

${\displaystyle{\displaystyle\exp\NVar{z}}}$

xml-id

C4.S2.E19.m2adec

0 references

base of natural logarithm

Defining formula

${\displaystyle{\displaystyle\mathrm{e}}}$

xml-id

C4.S2.E11.m2abdec

0 references

Q10755

Defining formula

${\displaystyle{\displaystyle!}}$

xml-id

introduction.Sx4.p1.t1.r15.m5aadec

0 references

hyperbolic cosine function

Defining formula

${\displaystyle{\displaystyle\cosh\NVar{z}}}$

xml-id

C4.S28.E2.m2adec

0 references

hyperbolic tangent function

Defining formula

${\displaystyle{\displaystyle\tanh\NVar{z}}}$

xml-id

C4.S28.E4.m2adec

0 references

imaginary unit

Defining formula

${\displaystyle{\displaystyle\mathrm{i}}}$

xml-id

C1.S9.E1.m2abdec

0 references

Q10771

Defining formula

${\displaystyle{\displaystyle\int}}$

xml-id

C1.S4.SS4.m3abdec

0 references

Q10816

Defining formula

${\displaystyle{\displaystyle\operatorname{arctan}\NVar{z}}}$

xml-id

C4.S23.SS2.p1.m7adec

0 references

irregular Coulomb radial functions

Defining formula

${\displaystyle{\displaystyle{H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)}}$

xml-id

C33.S2.E7.m2aadec

0 references

Q12666

Defining formula

${\displaystyle{\displaystyle\ell}}$

xml-id

C33.S1.XMD2.m1bdec

0 references

Q12667

Defining formula

${\displaystyle{\displaystyle\rho}}$

xml-id

C33.S1.XMD5.m1bdec

0 references

Q12668

Defining formula

${\displaystyle{\displaystyle\eta}}$

xml-id

C33.S1.XMD7.m1bdec

0 references

 

Sitelinks

Wikipedia(0 entries)

DRMF related sites(0 entries)

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