DLMF:36.9.E9
type: Item
Digital Library of Mathematical Functions ID: 36.9.E9
DLMF defining formula: <math xmlns="http://www.w3.org/1998/Math/MathML" id="p1.1.m1.1" class="ltx_Math" alttext="{\displaystyle{\displaystyle\left|\Psi^{(\mathrm{E})}\left(x,y,z\right)\right|% ^{2}=\frac{8\pi^{2}}{3^{2/3}}\int_{0}^{\infty}\int_{0}^{2\pi}\Re\left missing(% \mathrm{Ai}\left(\frac{1}{3^{1/3}}\left(x+iy+2zu\exp\left(i\theta\right)+3u^{2% }\exp\left(-2i\theta\right)\right)\right)\*\mathrm{Bi}\left(\frac{1}{3^{1/3}}% \left(x-iy+2zu\exp\left(-i\theta\right)+3u^{2}\exp\left(2i\theta\right)\right)% \right)\right)u\mathrm{d}u\mathrm{d}\theta.{}}}" display="inline"> | Ψ ( E ) ⁡ ( x , y , z ) | 2 = 8 ⁢ π 2 3 2 / 3 ⁢ ∫ 0 ∞ ∫ 0 2 ⁢ π ℜ ⁡ missing ⁢ ( Ai ⁡ ( 1 3 1 / 3 ⁢ ( x + i ⁢ y + 2 ⁢ z ⁢ u ⁢ exp ⁡ ( i ⁢ θ ) + 3 ⁢ u 2 ⁢ exp ⁡ ( - 2 ⁢ i ⁢ θ ) ) ) ⁢ Bi ⁡ ( 1 3 1 / 3 ⁢ ( x - i ⁢ y + 2 ⁢ z ⁢ u ⁢ exp ⁡ ( - i ⁢ θ ) + 3 ⁢ u 2 ⁢ exp ⁡ ( 2 ⁢ i ⁢ θ ) ) ) ) ⁢ u ⁢ d u ⁢ d θ . <annotation-xml encoding="MathML-Content" id="p1.1.m1.1b"> <apply id="p1.1.m1.1.23.2.cmml" xref="p1.1.m1.1.23"> <eq id="p1.1.m1.1.5.cmml" xref="p1.1.m1.1.5"/> <apply id="p1.1.m1.1.23.2.1.cmml" xref="p1.1.m1.1.23.2.1"> <csymbol cd="ambiguous" id="p1.1.m1.1.23.2.1.1.cmml" xref="p1.1.m1.1.23.2.1">superscript</csymbol> <apply id="p1.1.m1.1.23.2.1.2.1.cmml" xref="p1.1.m1.1.23.2.1.2"> <abs id="p1.1.m1.1.23.2.1.2.1.1.cmml" xref="p1.1.m1.1.1"/> <apply id="p1.1.m1.1.2.1.cmml" xref="p1.1.m1.1.2.7.2"> <csymbol cd="dlmf" id="p1.1.m1.1.2.2.1.1.cmml" xref="p1.1.m1.1.2.2.1.2">elliptic-umbilic-canonical-integral</csymbol> <ci id="p1.1.m1.1.2.5.1.cmml" xref="p1.1.m1.1.2.5.1">𝑥</ci> <ci id="p1.1.m1.1.2.5.3.cmml" xref="p1.1.m1.1.2.5.3">𝑦</ci> <ci id="p1.1.m1.1.2.5.5.cmml" xref="p1.1.m1.1.2.5.5">𝑧</ci> </apply> </apply> <cn type="integer" id="p1.1.m1.1.4.1.cmml" xref="p1.1.m1.1.4.1">2</cn> </apply> <apply id="p1.1.m1.1.23.2.2.cmml" xref="p1.1.m1.1.23.2.2"> <times id="p1.1.m1.1.23.2.2.1.cmml" xref="p1.1.m1.1.23.2.2.1"/> <apply id="p1.1.m1.1.6.cmml" xref="p1.1.m1.1.6"> <divide id="p1.1.m1.1.6.1.cmml" xref="p1.1.m1.1.6"/> <apply id="p1.1.m1.1.6.2.…
Symbols used: Q10893
Symbols used: Q11055
Symbols used: the ratio of the circumference of a circle to its diameter
Symbols used: differential of x
Symbols used: elliptic umbilic canonical integral function
Symbols used: exponential function
Symbols used: imaginary unit
Symbols used: integral
Symbols used: real part
Symbols used: real parameter
instance of: Digital Library of Mathematical Functions Formula
MaRDI profile type: MaRDI formula profile