DLMF:18.27.E20 (Q5971): Difference between revisions

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Revision as of 01:51, 2 January 2020 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Changed an Item: Add constraint ← Older edit		Latest revision as of 01:51, 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.m2adec
		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.m1abdec
		Property / Symbols used
			exponential function
		Property / Symbols used: exponential function / rank
			Normal rank
		Property / Symbols used: exponential function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\exp\NVar{z}}}$ \exp@@{\NVar{z}}
		Property / Symbols used: exponential function / qualifier
			xml-id: C4.S2.E19.m2adec
		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
			principal branch of logarithm function
		Property / Symbols used: principal branch of logarithm function / rank
			Normal rank
		Property / Symbols used: principal branch of logarithm function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\ln\NVar{z}}}$ \ln@@{\NVar{z}}
		Property / Symbols used: principal branch of logarithm function / qualifier
			xml-id: C4.S2.E2.m2aadec
		Property / Symbols used
			Q11298
		Property / Symbols used: Q11298 / rank
			Normal rank
		Property / Symbols used: Q11298 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}$ \qPochhammer{\NVar{a}}{\NVar{q}}{\NVar{n}}
		Property / Symbols used: Q11298 / qualifier
			xml-id: C17.S2.SS1.p1.m2afdec
		Property / Symbols used
			Q11728
		Property / Symbols used: Q11728 / rank
			Normal rank
		Property / Symbols used: Q11728 / qualifier
			Defining formula: ${\displaystyle{\displaystyle q}}$ q
		Property / Symbols used: Q11728 / qualifier
			xml-id: C18.S1.XMD3.m1rdec
		Property / Symbols used
			Q11730
		Property / Symbols used: Q11730 / rank
			Normal rank
		Property / Symbols used: Q11730 / qualifier
			Defining formula: ${\displaystyle{\displaystyle m}}$ m
		Property / Symbols used: Q11730 / qualifier
			xml-id: C18.S1.XMD5.m1fdec
		Property / Symbols used
			Q11726
		Property / Symbols used: Q11726 / rank
			Normal rank
		Property / Symbols used: Q11726 / qualifier
			Defining formula: ${\displaystyle{\displaystyle n}}$ n
		Property / Symbols used: Q11726 / qualifier
			xml-id: C18.S1.XMD6.m1odec
		Property / Symbols used
			Q11727
		Property / Symbols used: Q11727 / rank
			Normal rank
		Property / Symbols used: Q11727 / qualifier
			Defining formula: ${\displaystyle{\displaystyle x}}$ x
		Property / Symbols used: Q11727 / qualifier
			xml-id: C18.S2.XMD3.m1ndec

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Latest revision as of 01:51, 2 January 2020

No description defined

Language	Label	Description	Also known as
English	DLMF:18.27.E20	No description defined

Statements

Defining formula

${\displaystyle{\displaystyle\int_{0}^{\infty}S_{n}\left(q^{\frac{1}{2}}x;q\right)S_{m}\left(q^{\frac{1}{2}}x;q\right)\exp\left(-\frac{(\ln x)^{2}}{2\ln\left(q^{-1}\right)}\right)\mathrm{d}x=\frac{\sqrt{2\pi q^{-1}\ln\left(q^{-1}\right)}}{q^{n}\left(q;q\right)_{n}}\delta_{n,m}.}}$

0 references

DLMF id

DLMF:18.27.E20

0 references

Symbols used

Q10768

Defining formula

${\displaystyle{\displaystyle\delta_{\NVar{j},\NVar{k}}}}$

xml-id

introduction.Sx4.p1.t1.r4.m5afdec

0 references

Stieltjes–Wigert polynomial

Defining formula

${\displaystyle{\displaystyle S_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)}}$

xml-id

C18.S27.E18.m2abdec

0 references

the ratio of the circumference of a circle to its diameter

Defining formula

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

xml-id

C3.S12.E1.m2adec

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

Q10771

Defining formula

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

xml-id

C1.S4.SS4.m3abdec

0 references

principal branch of logarithm function

Defining formula

${\displaystyle{\displaystyle\ln\NVar{z}}}$

xml-id

C4.S2.E2.m2aadec

0 references

Q11298

Defining formula

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

xml-id

C17.S2.SS1.p1.m2afdec

0 references

Q11728

Defining formula

${\displaystyle{\displaystyle q}}$

xml-id

C18.S1.XMD3.m1rdec

0 references

Q11730

Defining formula

${\displaystyle{\displaystyle m}}$

xml-id

C18.S1.XMD5.m1fdec

0 references

Q11726

Defining formula

${\displaystyle{\displaystyle n}}$

xml-id

C18.S1.XMD6.m1odec

0 references

Q11727

Defining formula

${\displaystyle{\displaystyle x}}$

xml-id

C18.S2.XMD3.m1ndec

0 references

 

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