DLMF:28.28.E17 (Q8441): Difference between revisions

Latest revision as of 15:22, 2 January 2020

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Language	Label	Description	Also known as
English	DLMF:28.28.E17	No description defined

Statements

Defining formula

{\displaystyle{\displaystyle\dfrac{1}{\pi}\int_{0}^{\pi}\mathcal{C}^{(j)}_{\nu% +2s}(2hR)e^{-\mathrm{i}(\nu+2s)\phi}\mathrm{me}_{\nu}\left(t,h^{2}\right)% \mathrm{d}t=(-1)^{s}c^{\nu}_{2s}(h^{2}){\mathrm{M}^{(j)}_{\nu}}\left(z,h\right% ),}}

0 references

DLMF id

DLMF:28.28.E17

0 references

constraint

{\displaystyle{\displaystyle s\in\mathbb{Z}}}

0 references

Symbols used

Q12299

Defining formula

{\displaystyle{\displaystyle\mathrm{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}

xml-id

C28.S12.SS2.p2.m2acdec

0 references

the ratio of the circumference of a circle to its diameter

Defining formula

{\displaystyle{\displaystyle\pi}}

xml-id

C3.S12.E1.m2aodec

0 references

Q10770

Defining formula

{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

xml-id

C1.S4.SS4.m1amdec

0 references

Q10784

Defining formula

{\displaystyle{\displaystyle\in}}

xml-id

introduction.Sx4.p1.t1.r10.m2adec

0 references

base of natural logarithm

Defining formula

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

xml-id

C4.S2.E11.m2akdec

0 references

imaginary unit

Defining formula

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

xml-id

C1.S9.E1.m2aldec

0 references

Q10818

Defining formula

{\displaystyle{\displaystyle\mathbb{Z}}}

xml-id

introduction.Sx4.p2.t1.r20.m2adec

0 references

Q10771

Defining formula

{\displaystyle{\displaystyle\int}}

xml-id

C1.S4.SS4.m3amdec

0 references

Q12301

Defining formula

{\displaystyle{\displaystyle{\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{% z},\NVar{h}\right)}}

xml-id

C28.S20.SS3.p1.m4aedec

0 references

Q12294

Defining formula

{\displaystyle{\displaystyle h}}

xml-id

C28.S1.XMD11.m1ndec

0 references

Q12291

Defining formula

{\displaystyle{\displaystyle j}}

xml-id

C28.S1.XMD3.m1cdec

0 references

Q12272

Defining formula

{\displaystyle{\displaystyle z}}

xml-id

C28.S1.XMD6.m1mdec

0 references

Q12278

Defining formula

{\displaystyle{\displaystyle\nu}}

xml-id

C28.S1.XMD7.m1edec

0 references

Q12300

Defining formula

{\displaystyle{\displaystyle c_{2m}(q)}}

xml-id

C28.S14.XMD1.m1dec

0 references

Q12304

Defining formula

{\displaystyle{\displaystyle\mathcal{C}_{\mu}^{(j)}}}

xml-id

C28.S23.XMD1.m1dec

0 references

Q12314

Defining formula

{\displaystyle{\displaystyle\phi(z,t)}}

xml-id

C28.S28.XMD4.m1dec

0 references

Q12315

Defining formula

{\displaystyle{\displaystyle R(z,t)}}

xml-id

C28.S28.XMD5.m1dec

0 references

@@ Property / constraint @@
+ ${\displaystyle{\displaystyle s\in\mathbb{Z}}}$ 
s\in\Integers
@@ Property / constraint: ${\displaystyle{\displaystyle s\in\mathbb{Z}}}$ / rank @@
+Normal rank
@@ Property / Symbols used @@
+Q12299
@@ Property / Symbols used: Q12299 / rank @@
+Normal rank
@@ Property / Symbols used: Q12299 / qualifier @@
+Defining formula:  ${\displaystyle{\displaystyle\mathrm{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}$ 
\Mathieume{\NVar{n}}@{\NVar{z}}{\NVar{q}}
@@ Property / Symbols used: Q12299 / qualifier @@
+xml-id: C28.S12.SS2.p2.m2acdec
@@ Property / Symbols used @@
+the ratio of the circumference of a circle to its diameter
+Normal rank
+Defining formula:  ${\displaystyle{\displaystyle\pi}}$ 
\cpi
+xml-id: C3.S12.E1.m2aodec
@@ 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.m1amdec
@@ Property / Symbols used @@
+Q10784
@@ Property / Symbols used: Q10784 / rank @@
+Normal rank
@@ Property / Symbols used: Q10784 / qualifier @@
+Defining formula:  ${\displaystyle{\displaystyle\in}}$ 
\in
@@ Property / Symbols used: Q10784 / qualifier @@
+xml-id: introduction.Sx4.p1.t1.r10.m2adec
@@ 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.m2akdec
@@ 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.m2aldec
@@ Property / Symbols used @@
+Q10818
@@ Property / Symbols used: Q10818 / rank @@
+Normal rank
@@ Property / Symbols used: Q10818 / qualifier @@
+Defining formula:  ${\displaystyle{\displaystyle\mathbb{Z}}}$ 
\Integers
@@ Property / Symbols used: Q10818 / qualifier @@
+xml-id: introduction.Sx4.p2.t1.r20.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.m3amdec
@@ Property / Symbols used @@
+Q12301
@@ Property / Symbols used: Q12301 / rank @@
+Normal rank
@@ Property / Symbols used: Q12301 / qualifier @@
+Defining formula:  ${\displaystyle{\displaystyle{\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{% z},\NVar{h}\right)}}$ 
\modMathieuM{\NVar{j}}{\NVar{\nu}}@{\NVar{z}}{\NVar{h}}
@@ Property / Symbols used: Q12301 / qualifier @@
+xml-id: C28.S20.SS3.p1.m4aedec
@@ Property / Symbols used @@
+Q12294
@@ Property / Symbols used: Q12294 / rank @@
+Normal rank
@@ Property / Symbols used: Q12294 / qualifier @@
+Defining formula:  ${\displaystyle{\displaystyle h}}$ 
h
@@ Property / Symbols used: Q12294 / qualifier @@
+xml-id: C28.S1.XMD11.m1ndec
@@ Property / Symbols used @@
+Q12291
@@ Property / Symbols used: Q12291 / rank @@
+Normal rank
@@ Property / Symbols used: Q12291 / qualifier @@
+Defining formula:  ${\displaystyle{\displaystyle j}}$ 
j
@@ Property / Symbols used: Q12291 / qualifier @@
+xml-id: C28.S1.XMD3.m1cdec
@@ Property / Symbols used @@
+Q12272
@@ Property / Symbols used: Q12272 / rank @@
+Normal rank
@@ Property / Symbols used: Q12272 / qualifier @@
+Defining formula:  ${\displaystyle{\displaystyle z}}$ 
z
@@ Property / Symbols used: Q12272 / qualifier @@
+xml-id: C28.S1.XMD6.m1mdec
@@ Property / Symbols used @@
+Q12278
@@ Property / Symbols used: Q12278 / rank @@
+Normal rank
@@ Property / Symbols used: Q12278 / qualifier @@
+Defining formula:  ${\displaystyle{\displaystyle\nu}}$ 
\nu
@@ Property / Symbols used: Q12278 / qualifier @@
+xml-id: C28.S1.XMD7.m1edec
@@ Property / Symbols used @@
+Q12300
@@ Property / Symbols used: Q12300 / rank @@
+Normal rank
@@ Property / Symbols used: Q12300 / qualifier @@
+Defining formula:  ${\displaystyle{\displaystyle c_{2m}(q)}}$ 
c_{2m}(q)
@@ Property / Symbols used: Q12300 / qualifier @@
+xml-id: C28.S14.XMD1.m1dec
@@ Property / Symbols used @@
+Q12304
@@ Property / Symbols used: Q12304 / rank @@
+Normal rank
@@ Property / Symbols used: Q12304 / qualifier @@
+Defining formula:  ${\displaystyle{\displaystyle\mathcal{C}_{\mu}^{(j)}}}$ 
\mathcal{C}_{\mu}^{(j)}
@@ Property / Symbols used: Q12304 / qualifier @@
+xml-id: C28.S23.XMD1.m1dec
@@ Property / Symbols used @@
+Q12314
@@ Property / Symbols used: Q12314 / rank @@
+Normal rank
@@ Property / Symbols used: Q12314 / qualifier @@
+Defining formula:  ${\displaystyle{\displaystyle\phi(z,t)}}$ 
\phi(z,t)
@@ Property / Symbols used: Q12314 / qualifier @@
+xml-id: C28.S28.XMD4.m1dec
@@ Property / Symbols used @@
+Q12315
@@ Property / Symbols used: Q12315 / rank @@
+Normal rank
@@ Property / Symbols used: Q12315 / qualifier @@
+Defining formula:  ${\displaystyle{\displaystyle R(z,t)}}$ 
R(z,t)
@@ Property / Symbols used: Q12315 / qualifier @@
+xml-id: C28.S28.XMD5.m1dec

DLMF:28.28.E17 (Q8441): Difference between revisions

${\displaystyle{\displaystyle s\in\mathbb{Z}}}$

${\displaystyle{\displaystyle\mathrm{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}$

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

${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$

${\displaystyle{\displaystyle\in}}$

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

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

${\displaystyle{\displaystyle\mathbb{Z}}}$

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

${\displaystyle{\displaystyle{\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{% z},\NVar{h}\right)}}$

${\displaystyle{\displaystyle h}}$

${\displaystyle{\displaystyle j}}$

${\displaystyle{\displaystyle z}}$

${\displaystyle{\displaystyle\nu}}$

${\displaystyle{\displaystyle c_{2m}(q)}}$

${\displaystyle{\displaystyle\mathcal{C}_{\mu}^{(j)}}}$

${\displaystyle{\displaystyle\phi(z,t)}}$

${\displaystyle{\displaystyle R(z,t)}}$

Latest revision as of 15:22, 2 January 2020

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DLMF:28.28.E17 (Q8441): Difference between revisions

s ∈ ℤ 𝑠 {\displaystyle{\displaystyle s\in\mathbb{Z}}}

me n ⁡ ( z , q ) Mathieu-me 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}

π {\displaystyle{\displaystyle\pi}}

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

∈ {\displaystyle{\displaystyle\in}}

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

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

ℤ {\displaystyle{\displaystyle\mathbb{Z}}}

∫ {\displaystyle{\displaystyle\int}}

M ν ( j ) ⁡ ( z , h ) modified-Mathieu-M 𝑗 𝜈 𝑧 ℎ {\displaystyle{\displaystyle{\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{% z},\NVar{h}\right)}}

h ℎ {\displaystyle{\displaystyle h}}

j 𝑗 {\displaystyle{\displaystyle j}}

z 𝑧 {\displaystyle{\displaystyle z}}

ν 𝜈 {\displaystyle{\displaystyle\nu}}

c 2 ⁢ m ⁢ ( q ) subscript 𝑐 2 𝑚 𝑞 {\displaystyle{\displaystyle c_{2m}(q)}}

𝒞 μ ( j ) superscript subscript 𝒞 𝜇 𝑗 {\displaystyle{\displaystyle\mathcal{C}_{\mu}^{(j)}}}

ϕ ⁢ ( z , t ) italic-ϕ 𝑧 𝑡 {\displaystyle{\displaystyle\phi(z,t)}}

R ⁢ ( z , t ) 𝑅 𝑧 𝑡 {\displaystyle{\displaystyle R(z,t)}}

Latest revision as of 15:22, 2 January 2020

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${\displaystyle{\displaystyle s\in\mathbb{Z}}}$

${\displaystyle{\displaystyle\mathrm{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}$

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

${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$

${\displaystyle{\displaystyle\in}}$

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

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

${\displaystyle{\displaystyle\mathbb{Z}}}$

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

${\displaystyle{\displaystyle{\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{% z},\NVar{h}\right)}}$

${\displaystyle{\displaystyle h}}$

${\displaystyle{\displaystyle j}}$

${\displaystyle{\displaystyle z}}$

${\displaystyle{\displaystyle\nu}}$

${\displaystyle{\displaystyle c_{2m}(q)}}$

${\displaystyle{\displaystyle\mathcal{C}_{\mu}^{(j)}}}$

${\displaystyle{\displaystyle\phi(z,t)}}$

${\displaystyle{\displaystyle R(z,t)}}$