DLMF:22.10.E9 (Q7024)

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DLMF:22.10.E9
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    Statements

    Defining formula
    dn ⁡ ( z , k ) = sech ⁡ z + k ′ 2 4 ⁢ ( z + sinh ⁡ z ⁢ cosh ⁡ z ) ⁢ tanh ⁡ z ⁢ sech ⁡ z + O ⁡ ( k ′ 4 ) . Jacobi-elliptic-dn 𝑧 𝑘 𝑧 superscript superscript 𝑘 ′ 2 4 𝑧 𝑧 𝑧 𝑧 𝑧 Big-O superscript superscript 𝑘 ′ 4 {\displaystyle{\displaystyle\operatorname{dn}\left(z,k\right)=\operatorname{% sech}z+\frac{{k^{\prime}}^{2}}{4}(z+\sinh z\cosh z)\tanh z\operatorname{sech}z% +O\left({k^{\prime}}^{4}\right).}}
    0 references
    DLMF id
    DLMF:22.10.E9
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2ahdec
    0 references
    Jacobian elliptic function
    Defining formula
    dn ⁡ ( z , k ) Jacobi-elliptic-dn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{dn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E6.m2abdec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2abdec
    0 references
    hyperbolic secant function
    Defining formula
    sech ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{sech}\NVar{z}}}
    xml-id
    C4.S28.E6.m2abdec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2abdec
    0 references
    hyperbolic tangent function
    Defining formula
    tanh ⁡ z 𝑧 {\displaystyle{\displaystyle\tanh\NVar{z}}}
    xml-id
    C4.S28.E4.m2abdec
    0 references
    Q11985
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C22.S1.XMD3.m1hdec
    0 references
    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1hdec
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    Q11988
    Defining formula
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C22.S1.XMD5.m1bdec
    0 references
     
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