DLMF:36.2.E8
type: Item
Digital Library of Mathematical Functions ID: 36.2.E8
DLMF defining formula: <math xmlns="http://www.w3.org/1998/Math/MathML" id="p1.1.m1.1" class="ltx_Math" alttext="{\displaystyle{\displaystyle\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)=4\sqrt{% \ifrac{\pi}{6}}\,\exp\left(i\left(\tfrac{1}{27}z^{3}+\tfrac{1}{6}z(y+x)+\tfrac% {1}{4}\pi\right)\right)\*\int_{\infty\exp\left(5\pi i/12\right)}^{\infty\exp% \left(\pi i/12\right)}\exp\left(i\left(2u^{6}+2zu^{4}+\left(\tfrac{1}{2}z^{2}+% x+y\right)u^{2}-\frac{(y-x)^{2}}{24u^{2}}\right)\right)\mathrm{d}u,}}" display="inline"> Ψ ( H ) ⁡ ( 𝐱 ) = 4 ⁢ <mpadded width="+1.7pt" id="p1.1.m1.1.4" xref="p1.1.m1.1.4.cmml"> π / 6 </mpadded> ⁢ exp ⁡ ( i ⁢ ( 1 27 ⁢ z 3 + 1 6 ⁢ z ⁢ ( y + x ) + 1 4 ⁢ π ) ) ⁢ ∫ ∞ ⁢ exp ⁡ ( 5 ⁢ π ⁢ i / 12 ) ∞ ⁢ exp ⁡ ( π ⁢ i / 12 ) exp ⁡ ( i ⁢ ( 2 ⁢ u 6 + 2 ⁢ z ⁢ u 4 + ( 1 2 ⁢ z 2 + x + y ) ⁢ u 2 - ( y - x ) 2 24 ⁢ u 2 ) ) ⁢ d u , <annotation-xml encoding="MathML-Content" id="p1.1.m1.1b"> <apply id="p1.1.m1.1.14.2.cmml" xref="p1.1.m1.1.14"> <eq id="p1.1.m1.1.2.cmml" xref="p1.1.m1.1.2"/> <apply id="p1.1.m1.1.1.1.cmml" xref="p1.1.m1.1.1.7.2"> <csymbol cd="dlmf" id="p1.1.m1.1.1.2.1.1.cmml" xref="p1.1.m1.1.1.2.1.2">hyperbolic-umbilic-canonical-integral</csymbol> <ci id="p1.1.m1.1.1.5.cmml" xref="p1.1.m1.1.1.5">𝐱</ci> </apply> <apply id="p1.1.m1.1.14.2.1.cmml" xref="p1.1.m1.1.14.2.1"> <times id="p1.1.m1.1.7.cmml" xref="p1.1.m1.1.7"/> <cn type="integer" id="p1.1.m1.1.3.cmml" xref="p1.1.m1.1.3">4</cn> <apply id="p1.1.m1.1.4.cmml" xref="p1.1.m1.1.4"> <root id="p1.1.m1.1.4a.cmml" xref="p1.1.m1.1.4"/> <apply id="p1.1.m1.1.4.2.cmml" xref="p1.1.m1.1.4.2"> <divide id="p1.1.m1.1.4.2.1.1.cmml" xref="p1.1.m1.1.4.2.1.1"/> <ci id="p1.1.m1.1.4.2.1.2.cmml" xref="p1.1.m1.1.4.2.1.2">𝜋</ci> <cn type="integer" id="p1.1.m1.1.4.2.1.3.cmml" xref="p1.1.m1.1.4.2.1.3">6</cn> </apply> </apply> <apply id="p1.1.m1.1.6.1.cmml" xref="p1.1.m1.1.6.7.2"> <exp id="p1.1.m1.1.6.2.1.1.cmml" xref="p1.1.m1.1.6.2.1.2"/> <apply id="p1.1.m1.1.6.5.cmml" xref="p1.1.m1.1.6.5"> <times id="p1.1.m1.1.6.5.18.cmml" xref="p1.1.m1.1.6.5.18"…
Symbols used: the ratio of the circumference of a circle to its diameter
Symbols used: differential of x
Symbols used: exponential function
Symbols used: hyperbolic umbilic canonical integral function
Symbols used: imaginary unit
Symbols used: integral
Symbols used: real parameter
instance of: Digital Library of Mathematical Functions Formula
MaRDI profile type: MaRDI formula profile