DLMF:36.8.E3 (Q9947)

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DLMF:36.8.E3
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    Statements

    Defining formula
    3 2 / 3 4 ⁢ π 2 ⁢ Ψ ( H ) ⁡ ( 3 1 / 3 ⁢ 𝐱 ) = Ai ⁡ ( x ) ⁢ Ai ⁡ ( y ) ⁢ ∑ n = 0 ∞ ( - 3 - 1 / 3 ⁢ i ⁢ z ) n ⁢ c n ⁢ ( x ) ⁢ c n ⁢ ( y ) n ! + Ai ⁡ ( x ) ⁢ Ai ′ ⁡ ( y ) ⁢ ∑ n = 2 ∞ ( - 3 - 1 / 3 ⁢ i ⁢ z ) n ⁢ c n ⁢ ( x ) ⁢ d n ⁢ ( y ) n ! + Ai ′ ⁡ ( x ) ⁢ Ai ⁡ ( y ) ⁢ ∑ n = 2 ∞ ( - 3 - 1 / 3 ⁢ i ⁢ z ) n ⁢ d n ⁢ ( x ) ⁢ c n ⁢ ( y ) n ! + Ai ′ ⁡ ( x ) ⁢ Ai ′ ⁡ ( y ) ⁢ ∑ n = 1 ∞ ( - 3 - 1 / 3 ⁢ i ⁢ z ) n ⁢ d n ⁢ ( x ) ⁢ d n ⁢ ( y ) n ! , superscript 3 2 3 4 superscript 𝜋 2 hyperbolic-umbilic-canonical-integral superscript 3 1 3 𝐱 Airy-Ai 𝑥 Airy-Ai 𝑦 superscript subscript 𝑛 0 superscript superscript 3 1 3 𝑖 𝑧 𝑛 subscript 𝑐 𝑛 𝑥 subscript 𝑐 𝑛 𝑦 𝑛 Airy-Ai 𝑥 diffop Airy-Ai 1 𝑦 superscript subscript 𝑛 2 superscript superscript 3 1 3 𝑖 𝑧 𝑛 subscript 𝑐 𝑛 𝑥 subscript 𝑑 𝑛 𝑦 𝑛 diffop Airy-Ai 1 𝑥 Airy-Ai 𝑦 superscript subscript 𝑛 2 superscript superscript 3 1 3 𝑖 𝑧 𝑛 subscript 𝑑 𝑛 𝑥 subscript 𝑐 𝑛 𝑦 𝑛 diffop Airy-Ai 1 𝑥 diffop Airy-Ai 1 𝑦 superscript subscript 𝑛 1 superscript superscript 3 1 3 𝑖 𝑧 𝑛 subscript 𝑑 𝑛 𝑥 subscript 𝑑 𝑛 𝑦 𝑛 {\displaystyle{\displaystyle\dfrac{3^{2/3}}{4\pi^{2}}\Psi^{(\mathrm{H})}\left(% 3^{1/3}\mathbf{x}\right)=\mathrm{Ai}\left(x\right)\mathrm{Ai}\left(y\right)% \sum\limits_{n=0}^{\infty}(-3^{-1/3}iz)^{n}\dfrac{c_{n}(x)c_{n}(y)}{n!}+% \mathrm{Ai}\left(x\right)\mathrm{Ai}'\left(y\right)\sum\limits_{n=2}^{\infty}(% -3^{-1/3}iz)^{n}\dfrac{c_{n}(x)d_{n}(y)}{n!}+\mathrm{Ai}'\left(x\right)\mathrm% {Ai}\left(y\right)\sum\limits_{n=2}^{\infty}(-3^{-1/3}iz)^{n}\dfrac{d_{n}(x)c_% {n}(y)}{n!}+\mathrm{Ai}'\left(x\right)\mathrm{Ai}'\left(y\right)\sum\limits_{n% =1}^{\infty}(-3^{-1/3}iz)^{n}\dfrac{d_{n}(x)d_{n}(y)}{n!},}}
    0 references
    DLMF id
    DLMF:36.8.E3
    0 references
    Symbols used
    Q10919
    Defining formula
    Ai ⁡ ( z ) Airy-Ai 𝑧 {\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m1adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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    hyperbolic umbilic canonical integral function
    Defining formula
    Ψ ( H ) ⁡ ( 𝐱 ) hyperbolic-umbilic-canonical-integral 𝐱 {\displaystyle{\displaystyle\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)}}
    xml-id
    C36.S2.E5.m4adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
     
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