DLMF:12.9.E3 (Q4152): Difference between revisions

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Revision as of 17:44, 30 December 2019 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Changed an Item: Add constraint ← Older edit		Latest revision as of 15:16, 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.m2adec
		Property / Symbols used
			Q10759
		Property / Symbols used: Q10759 / rank
			Normal rank
		Property / Symbols used: Q10759 / qualifier
			Defining formula: ${\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}$ \Pochhammersym{\NVar{a}}{\NVar{n}}
		Property / Symbols used: Q10759 / qualifier
			xml-id: C5.S2.SS3.m1abdec
		Property / Symbols used
			Q10931
		Property / Symbols used: Q10931 / rank
			Normal rank
		Property / Symbols used: Q10931 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\sim}}$ \asympexp
		Property / Symbols used: Q10931 / qualifier
			xml-id: C2.S1.SS3.p1.m11abdec
		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.m2abdec
		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.m2abdec
		Property / Symbols used
			Q10755
		Property / Symbols used: Q10755 / rank
			Normal rank
		Property / Symbols used: Q10755 / qualifier
			Defining formula: ${\displaystyle{\displaystyle!}}$ !
		Property / Symbols used: Q10755 / qualifier
			xml-id: introduction.Sx4.p1.t1.r15.m5abdec
		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.m2adec
		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.m1aadec
		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.m1abdec
		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.m1bdec
		Property / Symbols used
			Q11526
		Property / Symbols used: Q11526 / rank
			Normal rank
		Property / Symbols used: Q11526 / qualifier
			Defining formula: ${\displaystyle{\displaystyle s}}$ s
		Property / Symbols used: Q11526 / qualifier
			xml-id: C12.S1.XMD5.m1bdec
		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.m1bdec
		Property / Symbols used
			Q11527
		Property / Symbols used: Q11527 / rank
			Normal rank
		Property / Symbols used: Q11527 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\delta}}$ \delta
		Property / Symbols used: Q11527 / qualifier
			xml-id: C12.S1.XMD8.m1bdec

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Latest revision as of 15:16, 2 January 2020

No description defined

Language	Label	Description	Also known as
English	DLMF:12.9.E3	No description defined

Statements

Defining formula

${\displaystyle{\displaystyle U\left(a,z\right)\sim e^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}\sum_{s=0}^{\infty}(-1)^{s}\frac{{\left(\frac{1}{2}+a\right)_{2s}}}{s!(2z^{2})^{s}}\pm i\frac{\sqrt{2\pi}}{\Gamma\left(\tfrac{1}{2}+a\right)}e^{\mp i\pi a}e^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}\sum_{s=0}^{\infty}\frac{{\left(\tfrac{1}{2}-a\right)_{2s}}}{s!(2z^{2})^{s}},}}$

0 references

DLMF id

DLMF:12.9.E3

0 references

constraint

${\displaystyle{\displaystyle\tfrac{1}{4}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{5}{4}\pi-\delta}}$

0 references

Symbols used

gamma function

Defining formula

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

xml-id

C5.S2.E1.m2adec

0 references

Q10759

Defining formula

${\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}$

xml-id

C5.S2.SS3.m1abdec

0 references

Q10931

Defining formula

${\displaystyle{\displaystyle\sim}}$

xml-id

C2.S1.SS3.p1.m11abdec

0 references

the ratio of the circumference of a circle to its diameter

Defining formula

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

xml-id

C3.S12.E1.m2abdec

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.m5abdec

0 references

imaginary unit

Defining formula

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

xml-id

C1.S9.E1.m2adec

0 references

Q11320

Defining formula

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

xml-id

C12.S2.SS1.p1.m1aadec

0 references

Q10817

Defining formula

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

xml-id

C1.S9.E7.m1abdec

0 references

Q11513

Defining formula

${\displaystyle{\displaystyle z}}$

xml-id

C12.S1.XMD3.m1bdec

0 references

Q11526

Defining formula

${\displaystyle{\displaystyle s}}$

xml-id

C12.S1.XMD5.m1bdec

0 references

Q11516

Defining formula

${\displaystyle{\displaystyle a}}$

xml-id

C12.S1.XMD6.m1bdec

0 references

Q11527

Defining formula

${\displaystyle{\displaystyle\delta}}$

xml-id

C12.S1.XMD8.m1bdec

0 references

 

Sitelinks

Wikipedia(0 entries)

DRMF related sites(0 entries)

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