DLMF:36.12.E11 (Q10003): Difference between revisions

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Revision as of 01:09, 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 15:21, 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
			Q10919
		Property / Symbols used: Q10919 / rank
			Normal rank
		Property / Symbols used: Q10919 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}$ \AiryAi@{\NVar{z}}
		Property / Symbols used: Q10919 / qualifier
			xml-id: C9.S2.SS1.m1adec
		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.m2aadec
		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
			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.m2abdec
		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.m2abdec
		Property / Symbols used
			Q12745
		Property / Symbols used: Q12745 / rank
			Normal rank
		Property / Symbols used: Q12745 / qualifier
			Defining formula: ${\displaystyle{\displaystyle y}}$ y
		Property / Symbols used: Q12745 / qualifier
			xml-id: C36.S1.XMD10.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.m1cdec
		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.m1bdec
		Property / Symbols used
			Q12760
		Property / Symbols used: Q12760 / rank
			Normal rank
		Property / Symbols used: Q12760 / qualifier
			Defining formula: ${\displaystyle{\displaystyle f(u,\mathbf{y})}}$ f(u,\mathbf{y})
		Property / Symbols used: Q12760 / qualifier
			xml-id: C36.S12.XMD3.m1fdec

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Latest revision as of 15:21, 2 January 2020

No description defined

Language	Label	Description	Also known as
English	DLMF:36.12.E11	No description defined

Statements

Defining formula

${\displaystyle{\displaystyle I(y,k)=\frac{\Delta^{1/4}\pi\sqrt{2}}{k^{1/3}}\exp\left(ik\widetilde{f}\right)\left(\left(\frac{g_{+}}{\sqrt{f_{+}^{\prime\prime}}}+\frac{g_{-}}{\sqrt{-f_{-}^{\prime\prime}}}\right)\mathrm{Ai}\left(-k^{2/3}\Delta\right)\left(1+O\left(\frac{1}{k}\right)\right)-i\left(\frac{g_{+}}{\sqrt{f_{+}^{\prime\prime}}}-\frac{g_{-}}{\sqrt{-f_{-}^{\prime\prime}}}\right)\frac{\mathrm{Ai}'\left(-k^{2/3}\Delta\right)}{k^{1/3}\Delta^{1/2}}\left(1+O\left(\frac{1}{k}\right)\right)\right),}}$

0 references

DLMF id

DLMF:36.12.E11

0 references

Symbols used

Q10919

Defining formula

${\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}$

xml-id

C9.S2.SS1.m1adec

0 references

order not exceeding

Defining formula

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

xml-id

C2.S1.E3.m2aadec

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

exponential function

Defining formula

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

xml-id

C4.S2.E19.m2abdec

0 references

imaginary unit

Defining formula

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

xml-id

C1.S9.E1.m2abdec

0 references

Q12745

Defining formula

${\displaystyle{\displaystyle y}}$

xml-id

C36.S1.XMD10.m1dec

0 references

Q12750

Defining formula

${\displaystyle{\displaystyle k}}$

xml-id

C36.S1.XMD4.m1cdec

0 references

oscillatory integral (locally)

Defining formula

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

xml-id

C36.S12.XMD1.m1bdec

0 references

Q12760

Defining formula

${\displaystyle{\displaystyle f(u,\mathbf{y})}}$

xml-id

C36.S12.XMD3.m1fdec

0 references

 

Sitelinks

Wikipedia(0 entries)

DRMF related sites(0 entries)

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