DLMF:22.14.E12 (Q7087)

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DLMF:22.14.E12
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    Statements

    Defining formula
    ∫ cs ⁡ ( x , k ) ⁢ d x = ln ⁡ ( ns ⁡ ( x , k ) - ds ⁡ ( x , k ) ) . Jacobi-elliptic-cs 𝑥 𝑘 𝑥 Jacobi-elliptic-ns 𝑥 𝑘 Jacobi-elliptic-ds 𝑥 𝑘 {\displaystyle{\displaystyle\int\operatorname{cs}\left(x,k\right)\mathrm{d}x=% \ln\left(\operatorname{ns}\left(x,k\right)-\operatorname{ds}\left(x,k\right)% \right).}}
    0 references
    DLMF id
    DLMF:22.14.E12
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    Symbols used
    Jacobian elliptic function
    Defining formula
    cs ⁡ ( z , k ) Jacobi-elliptic-cs 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{cs}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E9.m3abdec
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    Jacobian elliptic function
    Defining formula
    ds ⁡ ( z , k ) Jacobi-elliptic-ds 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{ds}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E7.m3abdec
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    Jacobian elliptic function
    Defining formula
    ns ⁡ ( z , k ) Jacobi-elliptic-ns 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{ns}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E4.m3abdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1akdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3akdec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2agdec
    0 references
    Q12003
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C22.S1.XMD1.m1kdec
    0 references
    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1kdec
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