DLMF:19.25.E22 (Q7222)

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DLMF:19.25.E22
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    Statements

    DLMF defining formula
    el3 ⁡ ( x , k c , p ) = el1 ⁡ ( x , k c ) + 1 3 ⁢ ( 1 - p ) ⁢ R J ⁡ ( r , r + k c 2 , r + 1 , r + p ) . Bulirsch-integral-el-3 𝑥 subscript 𝑘 𝑐 𝑝 Bulirsch-integral-el-1 𝑥 subscript 𝑘 𝑐 1 3 1 𝑝 Carlson-integral-RJ 𝑟 𝑟 superscript subscript 𝑘 𝑐 2 𝑟 1 𝑟 𝑝 {\displaystyle{\displaystyle\mathrm{el3}\left(x,k_{c},p\right)=\mathrm{el1}% \left(x,k_{c}\right)+\tfrac{1}{3}{(1-p)}R_{J}\left(r,r+k_{c}^{2},r+1,r+p\right% ).}}
    0 references
    Symbols used
    Bulirsch’s incomplete elliptic integral of the first kind
    DLMF defining formula
    el1 ⁡ ( x , k c ) Bulirsch-integral-el-1 𝑥 subscript 𝑘 𝑐 {\displaystyle{\displaystyle\mathrm{el1}\left(\NVar{x},\NVar{k_{c}}\right)}}
    xml-id
    C19.S2.E15.m1abdec
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    Bulirsch’s incomplete elliptic integral of the third kind
    DLMF defining formula
    el3 ⁡ ( x , k c , p ) Bulirsch-integral-el-3 𝑥 subscript 𝑘 𝑐 𝑝 {\displaystyle{\displaystyle\mathrm{el3}\left(\NVar{x},\NVar{k_{c}},\NVar{p}% \right)}}
    xml-id
    C19.S2.E16.m2adec
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    symmetric elliptic integral of third kind
    DLMF defining formula
    R J ⁡ ( x , y , z , p ) Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑝 {\displaystyle{\displaystyle R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}% \right)}}
    xml-id
    C19.S16.E2.m2aedec
    0 references
    real or complex modulus
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1udec
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    inverse
    DLMF defining formula
    r 𝑟 {\displaystyle{\displaystyle r}}
    xml-id
    C19.S25.XMD1.m1bdec
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    Identifiers

     
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