DLMF:31.10.E8 (Q9059)

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DLMF:31.10.E8
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    Statements

    Defining formula
    sin 2 ⁡ θ ⁢ ( ∂ 2 ⁡ 𝒦 ∂ ⁡ θ 2 + ( ( 1 - 2 ⁢ γ ) ⁢ tan ⁡ θ + 2 ⁢ ( δ + ϵ - 1 2 ) ⁢ cot ⁡ θ ) ⁢ ∂ ⁡ 𝒦 ∂ ⁡ θ - 4 ⁢ α ⁢ β ⁢ 𝒦 ) + ∂ 2 ⁡ 𝒦 ∂ ⁡ ϕ 2 + ( ( 1 - 2 ⁢ δ ) ⁢ cot ⁡ ϕ - ( 1 - 2 ⁢ ϵ ) ⁢ tan ⁡ ϕ ) ⁢ ∂ ⁡ 𝒦 ∂ ⁡ ϕ = 0 . 2 𝜃 partial-derivative 𝒦 𝜃 2 1 2 𝛾 𝜃 2 𝛿 italic-ϵ 1 2 𝜃 partial-derivative 𝒦 𝜃 4 𝛼 𝛽 𝒦 partial-derivative 𝒦 italic-ϕ 2 1 2 𝛿 italic-ϕ 1 2 italic-ϵ italic-ϕ partial-derivative 𝒦 italic-ϕ 0 {\displaystyle{\displaystyle{\sin^{2}}\theta\left(\frac{{\partial}^{2}\mathcal% {K}}{{\partial\theta}^{2}}+\left((1-2\gamma)\tan\theta+2(\delta+\epsilon-% \tfrac{1}{2})\cot\theta\right)\frac{\partial\mathcal{K}}{\partial\theta}-4% \alpha\beta\mathcal{K}\right)+\frac{{\partial}^{2}\mathcal{K}}{{\partial\phi}^% {2}}+\left((1-2\delta)\cot\phi-(1-2\epsilon)\tan\phi\right)\frac{\partial% \mathcal{K}}{\partial\phi}=0.}}
    0 references
    DLMF id
    DLMF:31.10.E8
    0 references
    Symbols used
    cotangent function
    Defining formula
    cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}
    xml-id
    C4.S14.E7.m2adec
    0 references
    partial derivative of $$f$$ with respect to $$x$$
    Defining formula
    ∂ ⁡ f ∂ ⁡ x partial-derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}
    xml-id
    C1.S5.E3.m4abdec
    0 references
    partial differential of $$x$$
    Defining formula
    ∂ ⁡ x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}
    xml-id
    C1.S5.E3.m2abdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    tangent function
    Defining formula
    tan ⁡ z 𝑧 {\displaystyle{\displaystyle\tan\NVar{z}}}
    xml-id
    C4.S14.E4.m2adec
    0 references
    Q12458
    Defining formula
    γ 𝛾 {\displaystyle{\displaystyle\gamma}}
    xml-id
    C31.S1.XMD10.m1cdec
    0 references
    Q12459
    Defining formula
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C31.S1.XMD11.m1cdec
    0 references
    Q12460
    Defining formula
    ϵ italic-ϵ {\displaystyle{\displaystyle\epsilon}}
    xml-id
    C31.S1.XMD12.m1cdec
    0 references
    Q12463
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C31.S1.XMD8.m1adec
    0 references
    Q12464
    Defining formula
    β 𝛽 {\displaystyle{\displaystyle\beta}}
    xml-id
    C31.S1.XMD9.m1adec
    0 references
    kernel (locally)
    Defining formula
    𝒦 ⁢ ( z , t ) 𝒦 𝑧 𝑡 {\displaystyle{\displaystyle\mathcal{K}(z,t)}}
    xml-id
    C31.S10.XMD4.m1cdec
    0 references
    Q12495
    Defining formula
    θ 𝜃 {\displaystyle{\displaystyle\theta}}
    xml-id
    C31.S10.XMD6.m1adec
    0 references
    Q12496
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C31.S10.XMD7.m1adec
    0 references
     
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