DLMF:36.11.E2
type: Item
Digital Library of Mathematical Functions ID: 36.11.E2
DLMF defining formula: <math xmlns="http://www.w3.org/1998/Math/MathML" id="p1.1.m1.1" class="ltx_Math" alttext="{\displaystyle{\displaystyle\Psi_{K}\left(\mathbf{x}\right)=\sqrt{2\pi}\sum% \limits_{j=1}^{j_{\max}(\mathbf{x})}\exp\left(i\left(\Phi_{K}\left(t_{j}(% \mathbf{x});\mathbf{x}\right)+\tfrac{1}{4}\pi(-1)^{j+K+1}\right)\right)\left|% \frac{{\partial}^{2}\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)}{{% \partial t}^{2}}\right|^{-1/2}(1+o\left(1\right)).}}" display="inline"> Ψ K ⁡ ( 𝐱 ) = 2 ⁢ π ⁢ <munderover id="p1.1.m1.1.18.2.1.2.1a" xref="p1.1.m1.1.18.2.1.2.1.cmml"> ∑ j = 1 j max ⁢ ( 𝐱 ) </munderover> exp ⁡ ( i ⁢ ( Φ K ⁡ ( t j ⁢ ( 𝐱 ) ; 𝐱 ) + 1 4 ⁢ π ⁢ ( - 1 ) j + K + 1 ) ) ⁢ | <mpadded width="-1.7pt" id="p1.1.m1.1.9.2.2.1.1" xref="p1.1.m1.1.9.1.cmml"> ∂ 2 </mpadded> ⁡ Φ K ⁡ ( t j ⁢ ( 𝐱 ) ; 𝐱 ) ∂ ⁡ t 2 | - 1 / 2 ⁢ ( 1 + o ⁡ ( 1 ) ) . <annotation-xml encoding="MathML-Content" id="p1.1.m1.1b"> <apply id="p1.1.m1.1.18.2.cmml" xref="p1.1.m1.1.18"> <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"> <apply id="p1.1.m1.1.1.2.1.1.cmml" xref="p1.1.m1.1.1.2.1.4"> <csymbol cd="dlmf" id="p1.1.m1.1.1.2.1.1.1.cmml" xref="p1.1.m1.1.1.2.1.2">canonical-integral</csymbol> <ci id="p1.1.m1.1.1.2.1.3.1.cmml" xref="p1.1.m1.1.1.2.1.3.1">𝐾</ci> </apply> <ci id="p1.1.m1.1.1.5.cmml" xref="p1.1.m1.1.1.5">𝐱</ci> </apply> <apply id="p1.1.m1.1.18.2.1.cmml" xref="p1.1.m1.1.18.2.1"> <times id="p1.1.m1.1.18.2.1.1.cmml" xref="p1.1.m1.1.18.2.1.1"/> <apply id="p1.1.m1.1.3.cmml" xref="p1.1.m1.1.3"> <root id="p1.1.m1.1.3a.cmml" xref="p1.1.m1.1.3"/> <apply id="p1.1.m1.1.3.2.cmml" xref="p1.1.m1.1.3.2"> <times id="p1.1.m1.1.3.2.3.cmml" xref="p1.1.m1.1.3.2.3"/> <cn type="integer" id="p1.1.m1.1.3.2.1.cmml" xref="p1.1.m1.1.3.2.1">2</cn> <ci id="p1.1.m1.1.3.2.2.cmml" xref="p1.1.m1.1.3.2.2">𝜋</ci> </apply> </apply> <apply id="p1.1.m1.1.18.2.1.2.cmml" xref="p1.1.m1.1.18.2.1.2"> <apply id="p1.1.m1.1.18.2.1.2.1.cmml" xref="p1.1.m1.1.18.2.1.2.1"> <csymbol cd="ambiguous" id="p…
Symbols used: canonical integral function
Symbols used: the ratio of the circumference of a circle to its diameter
Symbols used: cuspoid catastrophe of codimension $$K$$
Symbols used: exponential function
Symbols used: imaginary unit
Symbols used: order less than
Symbols used: partial derivative of $$f$$ with respect to $$x$$
Symbols used: partial differential of $$x$$
Symbols used: variable
Symbols used: codimension
Symbols used: solutions
instance of: Digital Library of Mathematical Functions Formula
MaRDI profile type: MaRDI formula profile