DLMF:10.43.E26 (Q3640): Difference between revisions

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Revision as of 14:03, 2 January 2020 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Changed an Item: Add constraint ← Older edit		Latest revision as of 14:03, 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
			gamma function
		Property / Symbols used: gamma function / rank
			Normal rank
		Property / Symbols used: gamma function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$ \EulerGamma@{\NVar{z}}
		Property / Symbols used: gamma function / qualifier
			xml-id: C5.S2.E1.m2agdec
		Property / Symbols used
			Q10770
		Property / Symbols used: Q10770 / rank
			Normal rank
		Property / Symbols used: Q10770 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$ \diff{\NVar{x}}
		Property / Symbols used: Q10770 / qualifier
			xml-id: C1.S4.SS4.m1audec
		Property / Symbols used
			Q10901
		Property / Symbols used: Q10901 / rank
			Normal rank
		Property / Symbols used: Q10901 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\Im}}$ \imagpart@@
		Property / Symbols used: Q10901 / qualifier
			xml-id: C1.S9.E2.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.m3audec
		Property / Symbols used
			modified Bessel function of the second kind
		Property / Symbols used: modified Bessel function of the second kind / rank
			Normal rank
		Property / Symbols used: modified Bessel function of the second kind / qualifier
			Defining formula: ${\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}$ \modBesselK{\NVar{\nu}}@{\NVar{z}}
		Property / Symbols used: modified Bessel function of the second kind / qualifier
			xml-id: C10.S25.E3.m2aldec
		Property / Symbols used
			Q10811
		Property / Symbols used: Q10811 / rank
			Normal rank
		Property / Symbols used: Q10811 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\Re}}$ \realpart@@
		Property / Symbols used: Q10811 / qualifier
			xml-id: C1.S9.E2.m1aidec
		Property / Symbols used
			$$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
		Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / rank
			Normal rank
		Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}$ \hyperOlverF@{\NVar{a}}{\NVar{b}}{\NVar{c}}{\NVar{z}}
		Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / qualifier
			xml-id: C15.S2.E2.m2adec
		Property / Symbols used
			Q11427
		Property / Symbols used: Q11427 / rank
			Normal rank
		Property / Symbols used: Q11427 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\nu}}$ \nu
		Property / Symbols used: Q11427 / qualifier
			xml-id: C10.S1.XMD7.m1mdec

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Latest revision as of 14:03, 2 January 2020

No description defined

Language	Label	Description	Also known as
English	DLMF:10.43.E26	No description defined

Statements

Defining formula

${\displaystyle{\displaystyle\int_{0}^{\infty}\frac{K_{\mu}\left(at\right)J_{\nu}\left(bt\right)}{t^{\lambda}}\mathrm{d}t=\frac{b^{\nu}\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\mu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\lambda-\frac{1}{2}\mu+\frac{1}{2}\right)}{2^{\lambda+1}a^{\nu-\lambda+1}}\*\mathbf{F}\left(\frac{\nu-\lambda+\mu+1}{2},\frac{\nu-\lambda-\mu+1}{2};\nu+1;-\frac{b^{2}}{a^{2}}\right),}}$

0 references

DLMF id

DLMF:10.43.E26

0 references

constraint

${\displaystyle{\displaystyle\Re(\nu+1-\lambda)>|\Re\mu|,\Re a>|\Im b|}}$

0 references

Symbols used

Bessel function of the first kind

Defining formula

${\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}$

xml-id

C10.S2.E2.m2adec

0 references

gamma function

Defining formula

${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$

xml-id

C5.S2.E1.m2agdec

0 references

Q10770

Defining formula

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

xml-id

C1.S4.SS4.m1audec

0 references

Q10901

Defining formula

${\displaystyle{\displaystyle\Im}}$

xml-id

C1.S9.E2.m2abdec

0 references

Q10771

Defining formula

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

xml-id

C1.S4.SS4.m3audec

0 references

modified Bessel function of the second kind

Defining formula

${\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}$

xml-id

C10.S25.E3.m2aldec

0 references

Q10811

Defining formula

${\displaystyle{\displaystyle\Re}}$

xml-id

C1.S9.E2.m1aidec

0 references

$$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function

Defining formula

${\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}$

xml-id

C15.S2.E2.m2adec

0 references

Q11427

Defining formula

${\displaystyle{\displaystyle\nu}}$

xml-id

C10.S1.XMD7.m1mdec

0 references

 

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