DLMF:36.12.E3 (Q9994): Difference between revisions

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Revision as of 01:08, 30 December 2019 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Created a new Item: Wikidata Toolkit example test item creation		Latest revision as of 15:19, 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
			order not exceeding
		Property / Symbols used: order not exceeding / rank
			Normal rank
		Property / Symbols used: order not exceeding / qualifier
			Defining formula: ${\displaystyle{\displaystyle O\left(\NVar{x}\right)}}$ \bigO@{\NVar{x}}
		Property / Symbols used: order not exceeding / qualifier
			xml-id: C2.S1.E3.m2adec
		Property / Symbols used
			Q10768
		Property / Symbols used: Q10768 / rank
			Normal rank
		Property / Symbols used: Q10768 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\delta_{\NVar{j},\NVar{k}}}}$ \Kroneckerdelta{\NVar{j}}{\NVar{k}}
		Property / Symbols used: Q10768 / qualifier
			xml-id: introduction.Sx4.p1.t1.r4.m5adec
		Property / Symbols used
			canonical integral function
		Property / Symbols used: canonical integral function / rank
			Normal rank
		Property / Symbols used: canonical integral function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)}}$ \canonint{\NVar{K}}@{\NVar{\mathbf{x}}}
		Property / Symbols used: canonical integral function / qualifier
			xml-id: C36.S2.E4.m2adec
		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.m2aadec
		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.m2aadec
		Property / Symbols used
			partial derivative of $$f$$ with respect to $$x$$
		Property / Symbols used: partial derivative of $$f$$ with respect to $$x$$ / rank
			Normal rank
		Property / Symbols used: partial derivative of $$f$$ with respect to $$x$$ / qualifier
			Defining formula: ${\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}$ \pderiv{\NVar{f}}{\NVar{x}}
		Property / Symbols used: partial derivative of $$f$$ with respect to $$x$$ / qualifier
			xml-id: C1.S5.E3.m4aadec
		Property / Symbols used
			partial differential of $$x$$
		Property / Symbols used: partial differential of $$x$$ / rank
			Normal rank
		Property / Symbols used: partial differential of $$x$$ / qualifier
			Defining formula: ${\displaystyle{\displaystyle\partial\NVar{x}}}$ \pdiff{\NVar{x}}
		Property / Symbols used: partial differential of $$x$$ / qualifier
			xml-id: C1.S5.E3.m2aadec
		Property / Symbols used
			Q12741
		Property / Symbols used: Q12741 / rank
			Normal rank
		Property / Symbols used: Q12741 / qualifier
			Defining formula: ${\displaystyle{\displaystyle n}}$ n
		Property / Symbols used: Q12741 / qualifier
			xml-id: C36.S1.XMD2.m1dec
		Property / Symbols used
			Q12750
		Property / Symbols used: Q12750 / rank
			Normal rank
		Property / Symbols used: Q12750 / qualifier
			Defining formula: ${\displaystyle{\displaystyle k}}$ k
		Property / Symbols used: Q12750 / qualifier
			xml-id: C36.S1.XMD4.m1adec
		Property / Symbols used
			Q12743
		Property / Symbols used: Q12743 / rank
			Normal rank
		Property / Symbols used: Q12743 / qualifier
			Defining formula: ${\displaystyle{\displaystyle K}}$ K
		Property / Symbols used: Q12743 / qualifier
			xml-id: C36.S1.XMD7.m1dec
		Property / Symbols used
			oscillatory integral (locally)
		Property / Symbols used: oscillatory integral (locally) / rank
			Normal rank
		Property / Symbols used: oscillatory integral (locally) / qualifier
			Defining formula: ${\displaystyle{\displaystyle I(\mathbf{y},k)}}$ I(\mathbf{y},k)
		Property / Symbols used: oscillatory integral (locally) / qualifier
			xml-id: C36.S12.XMD1.m1adec
		Property / Symbols used
			function (locally)
		Property / Symbols used: function (locally) / rank
			Normal rank
		Property / Symbols used: function (locally) / qualifier
			Defining formula: ${\displaystyle{\displaystyle a_{m}(\mathbf{y})}}$ a_{m}(\mathbf{y})
		Property / Symbols used: function (locally) / qualifier
			xml-id: C36.S12.XMD10.m1dec
		Property / Symbols used
			function (locally)
		Property / Symbols used: function (locally) / rank
			Normal rank
		Property / Symbols used: function (locally) / qualifier
			Defining formula: ${\displaystyle{\displaystyle A(\mathbf{y})}}$ A(\mathbf{y})
		Property / Symbols used: function (locally) / qualifier
			xml-id: C36.S12.XMD7.m1dec
		Property / Symbols used
			Q12761
		Property / Symbols used: Q12761 / rank
			Normal rank
		Property / Symbols used: Q12761 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\mathbf{z}(\mathbf{y},k)}}$ \mathbf{z}(\mathbf{y},k)
		Property / Symbols used: Q12761 / qualifier
			xml-id: C36.S12.XMD8.m1dec
		Property / Symbols used
			Q12762
		Property / Symbols used: Q12762 / rank
			Normal rank
		Property / Symbols used: Q12762 / qualifier
			Defining formula: ${\displaystyle{\displaystyle z_{m}(\mathbf{y},k)}}$ z_{m}(\mathbf{y},k)
		Property / Symbols used: Q12762 / qualifier
			xml-id: C36.S12.XMD9.m1dec

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Latest revision as of 15:19, 2 January 2020

No description defined

Language	Label	Description	Also known as
English	DLMF:36.12.E3	No description defined

Statements

Defining formula

${\displaystyle{\displaystyle I(\mathbf{y},k)=\frac{\exp\left(ikA(\mathbf{y})\right)}{k^{1/(K+2)}}\sum\limits_{m=0}^{K}\frac{a_{m}(\mathbf{y})}{k^{m/(K+2)}}\left(\delta_{m,0}-\left(1-\delta_{m,0}\right)i\frac{\partial}{\partial z_{m}}\right)\Psi_{K}\left(\mathbf{z}(\mathbf{y};k)\right)\left(1+O\left(\frac{1}{k}\right)\right),}}$

0 references

DLMF id

DLMF:36.12.E3

0 references

Symbols used

order not exceeding

Defining formula

${\displaystyle{\displaystyle O\left(\NVar{x}\right)}}$

xml-id

C2.S1.E3.m2adec

0 references

Q10768

Defining formula

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

xml-id

introduction.Sx4.p1.t1.r4.m5adec

0 references

canonical integral function

Defining formula

${\displaystyle{\displaystyle\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)}}$

xml-id

C36.S2.E4.m2adec

0 references

exponential function

Defining formula

${\displaystyle{\displaystyle\exp\NVar{z}}}$

xml-id

C4.S2.E19.m2aadec

0 references

imaginary unit

Defining formula

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

xml-id

C1.S9.E1.m2aadec

0 references

partial derivative of $$f$$ with respect to $$x$$

Defining formula

${\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}$

xml-id

C1.S5.E3.m4aadec

0 references

partial differential of $$x$$

Defining formula

${\displaystyle{\displaystyle\partial\NVar{x}}}$

xml-id

C1.S5.E3.m2aadec

0 references

Q12741

Defining formula

${\displaystyle{\displaystyle n}}$

xml-id

C36.S1.XMD2.m1dec

0 references

Q12750

Defining formula

${\displaystyle{\displaystyle k}}$

xml-id

C36.S1.XMD4.m1adec

0 references

Q12743

Defining formula

${\displaystyle{\displaystyle K}}$

xml-id

C36.S1.XMD7.m1dec

0 references

oscillatory integral (locally)

Defining formula

${\displaystyle{\displaystyle I(\mathbf{y},k)}}$

xml-id

C36.S12.XMD1.m1adec

0 references

function (locally)

Defining formula

${\displaystyle{\displaystyle a_{m}(\mathbf{y})}}$

xml-id

C36.S12.XMD10.m1dec

0 references

function (locally)

Defining formula

${\displaystyle{\displaystyle A(\mathbf{y})}}$

xml-id

C36.S12.XMD7.m1dec

0 references

Q12761

Defining formula

${\displaystyle{\displaystyle\mathbf{z}(\mathbf{y},k)}}$

xml-id

C36.S12.XMD8.m1dec

0 references

Q12762

Defining formula

${\displaystyle{\displaystyle z_{m}(\mathbf{y},k)}}$

xml-id

C36.S12.XMD9.m1dec

0 references

 

Sitelinks

Wikipedia(0 entries)

DRMF related sites(0 entries)

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