DLMF:14.5.E26 (Q4738)

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DLMF:14.5.E26
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    Statements

    test
    𝑸 1 2 ⁑ ( cosh ⁑ ΞΎ ) = 2 ⁒ Ο€ - 1 / 2 ⁒ cosh ⁑ ΞΎ ⁒ sech ⁑ ( 1 2 ⁒ ΞΎ ) ⁒ K ⁑ ( sech ⁑ ( 1 2 ⁒ ΞΎ ) ) - 4 ⁒ Ο€ - 1 / 2 ⁒ cosh ⁑ ( 1 2 ⁒ ΞΎ ) ⁒ E ⁑ ( sech ⁑ ( 1 2 ⁒ ΞΎ ) ) , shorthand-associated-Legendre-black-Q 1 2 πœ‰ 2 superscript πœ‹ 1 2 πœ‰ 1 2 πœ‰ complete-elliptic-integral-first-kind-K 1 2 πœ‰ 4 superscript πœ‹ 1 2 1 2 πœ‰ complete-elliptic-integral-second-kind-E 1 2 πœ‰ {\displaystyle{\displaystyle\boldsymbol{Q}_{\frac{1}{2}}\left(\cosh\xi\right)=% 2\pi^{-1/2}\cosh\xi\operatorname{sech}\left(\tfrac{1}{2}\xi\right)K\left(% \operatorname{sech}\left(\tfrac{1}{2}\xi\right)\right)-4\pi^{-1/2}\cosh\left(% \tfrac{1}{2}\xi\right)E\left(\operatorname{sech}\left(\tfrac{1}{2}\xi\right)% \right),}}
    0 references
    DLMF id
    DLMF:14.5.E26
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    test
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aodec
    0 references
    Q11600
    test
    K ⁑ ( k ) complete-elliptic-integral-first-kind-K π‘˜ {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1aedec
    0 references
    Q11601
    test
    E ⁑ ( k ) complete-elliptic-integral-second-kind-E π‘˜ {\displaystyle{\displaystyle E\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m2acdec
    0 references
    hyperbolic cosine function
    test
    cosh ⁑ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2afdec
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    hyperbolic secant function
    test
    sech ⁑ z 𝑧 {\displaystyle{\displaystyle\operatorname{sech}\NVar{z}}}
    xml-id
    C4.S28.E6.m2adec
    0 references
    Q11597
    test
    𝑸 Ξ½ ⁑ ( z ) = 𝑸 Ξ½ 0 ⁑ ( z ) shorthand-associated-Legendre-black-Q 𝜈 𝑧 associated-Legendre-black-Q 0 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=% \boldsymbol{Q}^{0}_{\nu}\left(z\right)}}
    xml-id
    C14.S2.SS2.p2.m6abdec
    0 references
    Q11599
    test
    ΞΎ > 0 πœ‰ 0 {\displaystyle{\displaystyle\xi>0}}
    xml-id
    C14.S5.XMD1.m1fdec
    0 references
     
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