DLMF:4.23.E40 (Q1792)

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DLMF:4.23.E40
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    Statements

    test
    gd ⁡ ( x ) = 2 ⁢ arctan ⁡ ( e x ) - 1 2 ⁢ π = arcsin ⁡ ( tanh ⁡ x ) = arccsc ⁡ ( coth ⁡ x ) = arccos ⁡ ( sech ⁡ x ) = arcsec ⁡ ( cosh ⁡ x ) = arctan ⁡ ( sinh ⁡ x ) = arccot ⁡ ( csch ⁡ x ) . Gudermannian 𝑥 2 superscript 𝑒 𝑥 1 2 𝜋 𝑥 hyperbolic-cotangent 𝑥 𝑥 𝑥 𝑥 𝑥 {\displaystyle{\displaystyle\operatorname{gd}\left(x\right)=2\operatorname{% arctan}\left(e^{x}\right)-\tfrac{1}{2}\pi\\ =\operatorname{arcsin}\left(\tanh x\right)=\operatorname{arccsc}\left(\coth x% \right)\\ =\operatorname{arccos}\left(\operatorname{sech}x\right)=\operatorname{arcsec}% \left(\cosh x\right)\\ =\operatorname{arctan}\left(\sinh x\right)=\operatorname{arccot}\left(% \operatorname{csch}x\right).}}
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    DLMF id
    DLMF:4.23.E40
    0 references
    Symbols used
    Gudermannian function
    test
    gd ⁡ x Gudermannian 𝑥 {\displaystyle{\displaystyle\operatorname{gd}\NVar{x}}}
    xml-id
    C4.S23.E39.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2andec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    hyperbolic cosecant function
    test
    csch ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{csch}\NVar{z}}}
    xml-id
    C4.S28.E5.m2adec
    0 references
    hyperbolic cosine function
    test
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2adec
    0 references
    hyperbolic cotangent function
    test
    coth ⁡ z hyperbolic-cotangent 𝑧 {\displaystyle{\displaystyle\coth\NVar{z}}}
    xml-id
    C4.S28.E7.m2adec
    0 references
    hyperbolic secant function
    test
    sech ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{sech}\NVar{z}}}
    xml-id
    C4.S28.E6.m2aadec
    0 references
    hyperbolic sine function
    test
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
    0 references
    hyperbolic tangent function
    test
    tanh ⁡ z 𝑧 {\displaystyle{\displaystyle\tanh\NVar{z}}}
    xml-id
    C4.S28.E4.m2adec
    0 references
    arccosecant function
    test
    arccsc ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arccsc}\NVar{z}}}
    xml-id
    C4.S23.E7.m2aedec
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    Q11208
    test
    arccos ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arccos}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m6akdec
    0 references
    arccotangent function
    test
    arccot ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arccot}\NVar{z}}}
    xml-id
    C4.S23.E9.m2aedec
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    arcsecant function
    test
    arcsec ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arcsec}\NVar{z}}}
    xml-id
    C4.S23.E8.m2aedec
    0 references
    Q11242
    test
    arcsin ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arcsin}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m5ajdec
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    Q10816
    test
    arctan ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arctan}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m7aidec
    0 references
    Q11218
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C4.S1.XMD6.m1kdec
    0 references
     
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