DLMF:9.7.E13 (Q2820): Difference between revisions

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Revision as of 13:35, 29 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 14:35, 2 January 2020 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Changed an Item: Add constraint
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		Property / constraint
			${\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta}}$ |\phase@@{z}|\leq\tfrac{2}{3}\pi-\delta
		Property / constraint: ${\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta}}$ / rank
			Normal rank
		Property / Symbols used
			Q11081
		Property / Symbols used: Q11081 / rank
			Normal rank
		Property / Symbols used: Q11081 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\mathrm{Bi}\left(\NVar{z}\right)}}$ \AiryBi@{\NVar{z}}
		Property / Symbols used: Q11081 / qualifier
			xml-id: C9.S2.SS1.m2addec
		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.m11ahdec
		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.m2ajdec
		Property / Symbols used
			cosine function
		Property / Symbols used: cosine function / rank
			Normal rank
		Property / Symbols used: cosine function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\cos\NVar{z}}}$ \cos@@{\NVar{z}}
		Property / Symbols used: cosine function / qualifier
			xml-id: C4.S14.E2.m2addec
		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.m2addec
		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
			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.m2adec
		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.m1ahdec
		Property / Symbols used
			sine function
		Property / Symbols used: sine function / rank
			Normal rank
		Property / Symbols used: sine function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\sin\NVar{z}}}$ \sin@@{\NVar{z}}
		Property / Symbols used: sine function / qualifier
			xml-id: C4.S14.E1.m2addec
		Property / Symbols used
			Q11385
		Property / Symbols used: Q11385 / rank
			Normal rank
		Property / Symbols used: Q11385 / qualifier
			Defining formula: ${\displaystyle{\displaystyle k}}$ k
		Property / Symbols used: Q11385 / qualifier
			xml-id: C9.S1.XMD1.m1idec
		Property / Symbols used
			Q11381
		Property / Symbols used: Q11381 / rank
			Normal rank
		Property / Symbols used: Q11381 / qualifier
			Defining formula: ${\displaystyle{\displaystyle z}}$ z
		Property / Symbols used: Q11381 / qualifier
			xml-id: C9.S1.XMD3.m1idec
		Property / Symbols used
			Q11386
		Property / Symbols used: Q11386 / rank
			Normal rank
		Property / Symbols used: Q11386 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\delta}}$ \delta
		Property / Symbols used: Q11386 / qualifier
			xml-id: C9.S1.XMD4.m1hdec
		Property / Symbols used
			change of variable (locally)
		Property / Symbols used: change of variable (locally) / rank
			Normal rank
		Property / Symbols used: change of variable (locally) / qualifier
			Defining formula: ${\displaystyle{\displaystyle\zeta}}$ \zeta
		Property / Symbols used: change of variable (locally) / qualifier
			xml-id: C9.S7.XMD1.m1idec
		Property / Symbols used
			Q11387
		Property / Symbols used: Q11387 / rank
			Normal rank
		Property / Symbols used: Q11387 / qualifier
			Defining formula: ${\displaystyle{\displaystyle u_{s}}}$ u_{s}
		Property / Symbols used: Q11387 / qualifier
			xml-id: C9.S7.XMD2.m1edec

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Latest revision as of 14:35, 2 January 2020

No description defined

Language	Label	Description	Also known as
English	DLMF:9.7.E13	No description defined

Statements

Defining formula

${\displaystyle{\displaystyle\mathrm{Bi}\left(ze^{\pm\pi i/3}\right)\mathrel{\sim}\sqrt{\frac{2}{\pi}}\frac{e^{\pm\pi i/6}}{z^{1/4}}\*\left(\cos\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}\mathrm{i}\ln 2\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{u_{2k}}{\zeta^{2k}}+\sin\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}\mathrm{i}\ln 2\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{u_{2k+1}}{\zeta^{2k+1}}\right),}}$

0 references

DLMF id

DLMF:9.7.E13

0 references

constraint

${\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta}}$

0 references

Symbols used

Q11081

Defining formula

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

xml-id

C9.S2.SS1.m2addec

0 references

Q10931

Defining formula

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

xml-id

C2.S1.SS3.p1.m11ahdec

0 references

the ratio of the circumference of a circle to its diameter

Defining formula

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

xml-id

C3.S12.E1.m2ajdec

0 references

cosine function

Defining formula

${\displaystyle{\displaystyle\cos\NVar{z}}}$

xml-id

C4.S14.E2.m2addec

0 references

base of natural logarithm

Defining formula

${\displaystyle{\displaystyle\mathrm{e}}}$

xml-id

C4.S2.E11.m2addec

0 references

imaginary unit

Defining formula

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

xml-id

C1.S9.E1.m2adec

0 references

principal branch of logarithm function

Defining formula

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

xml-id

C4.S2.E2.m2adec

0 references

Q10817

Defining formula

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

xml-id

C1.S9.E7.m1ahdec

0 references

sine function

Defining formula

${\displaystyle{\displaystyle\sin\NVar{z}}}$

xml-id

C4.S14.E1.m2addec

0 references

Q11385

Defining formula

${\displaystyle{\displaystyle k}}$

xml-id

C9.S1.XMD1.m1idec

0 references

Q11381

Defining formula

${\displaystyle{\displaystyle z}}$

xml-id

C9.S1.XMD3.m1idec

0 references

Q11386

Defining formula

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

xml-id

C9.S1.XMD4.m1hdec

0 references

change of variable (locally)

Defining formula

${\displaystyle{\displaystyle\zeta}}$

xml-id

C9.S7.XMD1.m1idec

0 references

Q11387

Defining formula

${\displaystyle{\displaystyle u_{s}}}$

xml-id

C9.S7.XMD2.m1edec

0 references

 

Sitelinks

Wikipedia(0 entries)

DRMF related sites(0 entries)

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