DLMF:4.35.E37 (Q1911)

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DLMF:4.35.E37
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    Statements

    Defining formula
    coth ⁡ z = sinh ⁡ ( 2 ⁢ x ) - i ⁢ sin ⁡ ( 2 ⁢ y ) cosh ⁡ ( 2 ⁢ x ) - cos ⁡ ( 2 ⁢ y ) . hyperbolic-cotangent 𝑧 2 𝑥 𝑖 2 𝑦 2 𝑥 2 𝑦 {\displaystyle{\displaystyle\coth z=\frac{\sinh\left(2x\right)-i\sin\left(2y% \right)}{\cosh\left(2x\right)-\cos\left(2y\right)}.}}
    0 references
    DLMF id
    DLMF:4.35.E37
    0 references
    Symbols used
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2azdec
    0 references
    hyperbolic cotangent function
    Defining formula
    coth ⁡ z hyperbolic-cotangent 𝑧 {\displaystyle{\displaystyle\coth\NVar{z}}}
    xml-id
    C4.S28.E7.m2acdec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2aydec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2acdec
    0 references
    Q11218
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C4.S1.XMD6.m1cdec
    0 references
    Q11222
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C4.S1.XMD7.m1cdec
    0 references
    Q11217
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C4.S1.XMD8.m1tdec
    0 references
     
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