DLMF:10.32.E8 (Q3528): Difference between revisions

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Revision as of 13:42, 2 January 2020 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Changed an Item: Add constraint ← Older edit		Latest revision as of 13:42, 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
			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.m1abdec
		Property / Symbols used
			Q11426
		Property / Symbols used: Q11426 / rank
			Normal rank
		Property / Symbols used: Q11426 / qualifier
			Defining formula: ${\displaystyle{\displaystyle z}}$ z
		Property / Symbols used: Q11426 / qualifier
			xml-id: C10.S1.XMD6.m1edec
		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.m1cdec

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Latest revision as of 13:42, 2 January 2020

No description defined

Language	Label	Description	Also known as
English	DLMF:10.32.E8	No description defined

Statements

Defining formula

${\displaystyle{\displaystyle K_{\nu}\left(z\right)=\frac{\pi^{\frac{1}{2}}(\frac{1}{2}z)^{\nu}}{\Gamma\left(\nu+\frac{1}{2}\right)}\int_{0}^{\infty}e^{-z\cosh t}(\sinh t)^{2\nu}\mathrm{d}t=\frac{\pi^{\frac{1}{2}}(\frac{1}{2}z)^{\nu}}{\Gamma\left(\nu+\frac{1}{2}\right)}\int_{1}^{\infty}e^{-zt}(t^{2}-1)^{\nu-\frac{1}{2}}\mathrm{d}t,}}$

0 references

DLMF id

DLMF:10.32.E8

0 references

constraint

${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}}$

0 references

${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi}}$

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Symbols used

gamma function

Defining formula

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

xml-id

C5.S2.E1.m2aadec

0 references

the ratio of the circumference of a circle to its diameter

Defining formula

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

xml-id

C3.S12.E1.m2afdec

0 references

Q10770

Defining formula

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

xml-id

C1.S4.SS4.m1agdec

0 references

base of natural logarithm

Defining formula

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

xml-id

C4.S2.E11.m2aedec

0 references

hyperbolic cosine function

Defining formula

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

xml-id

C4.S28.E2.m2acdec

0 references

hyperbolic sine function

Defining formula

${\displaystyle{\displaystyle\sinh\NVar{z}}}$

xml-id

C4.S28.E1.m2abdec

0 references

Q10771

Defining formula

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

xml-id

C1.S4.SS4.m3agdec

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.m2acdec

0 references

Q10817

Defining formula

${\displaystyle{\displaystyle\operatorname{ph}}}$

xml-id

C1.S9.E7.m1aadec

0 references

Q10811

Defining formula

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

xml-id

C1.S9.E2.m1abdec

0 references

Q11426

Defining formula

${\displaystyle{\displaystyle z}}$

xml-id

C10.S1.XMD6.m1edec

0 references

Q11427

Defining formula

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

xml-id

C10.S1.XMD7.m1cdec

0 references

 

Sitelinks

Wikipedia(0 entries)

DRMF related sites(0 entries)

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