DLMF:12.5.E8 (Q4122): Difference between revisions

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Revision as of 15:10, 2 January 2020 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Changed an Item: Add constraint ← Older edit		Latest revision as of 15:10, 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
			the ratio of the circumference of a circle to its diameter
		Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
			Normal rank
		Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
			Defining formula: ${\displaystyle{\displaystyle\pi}}$ \cpi
		Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
			xml-id: C3.S12.E1.m2afdec
		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.m1agdec
		Property / Symbols used
			base of natural logarithm
		Property / Symbols used: base of natural logarithm / rank
			Normal rank
		Property / Symbols used: base of natural logarithm / qualifier
			Defining formula: ${\displaystyle{\displaystyle\mathrm{e}}}$ \expe
		Property / Symbols used: base of natural logarithm / qualifier
			xml-id: C4.S2.E11.m2agdec
		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.m2acdec
		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.m3agdec
		Property / Symbols used
			Q11320
		Property / Symbols used: Q11320 / rank
			Normal rank
		Property / Symbols used: Q11320 / qualifier
			Defining formula: ${\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}$ \paraU@{\NVar{a}}{\NVar{z}}
		Property / Symbols used: Q11320 / qualifier
			xml-id: C12.S2.SS1.p1.m1afdec
		Property / Symbols used
			Q10817
		Property / Symbols used: Q10817 / rank
			Normal rank
		Property / Symbols used: Q10817 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\operatorname{ph}}}$ \phase
		Property / Symbols used: Q10817 / qualifier
			xml-id: C1.S9.E7.m1aedec
		Property / Symbols used
			Q11513
		Property / Symbols used: Q11513 / rank
			Normal rank
		Property / Symbols used: Q11513 / qualifier
			Defining formula: ${\displaystyle{\displaystyle z}}$ z
		Property / Symbols used: Q11513 / qualifier
			xml-id: C12.S1.XMD3.m1gdec
		Property / Symbols used
			Q11516
		Property / Symbols used: Q11516 / rank
			Normal rank
		Property / Symbols used: Q11516 / qualifier
			Defining formula: ${\displaystyle{\displaystyle a}}$ a
		Property / Symbols used: Q11516 / qualifier
			xml-id: C12.S1.XMD6.m1gdec

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Latest revision as of 15:10, 2 January 2020

No description defined

Language	Label	Description	Also known as
English	DLMF:12.5.E8	No description defined

Statements

Defining formula

${\displaystyle{\displaystyle U\left(a,z\right)=\frac{e^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}}{2\pi i\Gamma\left(\frac{1}{2}+a\right)}\*\int_{-i\infty}^{i\infty}\Gamma\left(t\right)\Gamma\left(\tfrac{1}{2}+a-2t\right)2^{t}z^{2t}\mathrm{d}t,}}$

0 references

DLMF id

DLMF:12.5.E8

0 references

constraint

${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{3}{4}\pi}}$

0 references

${\displaystyle{\displaystyle a\neq-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\dots}}$

0 references

${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{3}{4}\pi}}$

0 references

Symbols used

gamma function

Defining formula

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

xml-id

C5.S2.E1.m2addec

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.m2agdec

0 references

imaginary unit

Defining formula

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

xml-id

C1.S9.E1.m2acdec

0 references

Q10771

Defining formula

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

xml-id

C1.S4.SS4.m3agdec

0 references

Q11320

Defining formula

${\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}$

xml-id

C12.S2.SS1.p1.m1afdec

0 references

Q10817

Defining formula

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

xml-id

C1.S9.E7.m1aedec

0 references

Q11513

Defining formula

${\displaystyle{\displaystyle z}}$

xml-id

C12.S1.XMD3.m1gdec

0 references

Q11516

Defining formula

${\displaystyle{\displaystyle a}}$

xml-id

C12.S1.XMD6.m1gdec

0 references

 

Sitelinks

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DRMF related sites(0 entries)

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