DLMF:4.23.E42 (Q1794)

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DLMF:4.23.E42
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    Statements

    Defining formula
    gd - 1 ⁡ ( x ) = ln ⁡ tan ⁡ ( 1 2 ⁢ x + 1 4 ⁢ π ) = ln ⁡ ( sec ⁡ x + tan ⁡ x ) = arcsinh ⁡ ( tan ⁡ x ) = arccsch ⁡ ( cot ⁡ x ) = arccosh ⁡ ( sec ⁡ x ) = arcsech ⁡ ( cos ⁡ x ) = arctanh ⁡ ( sin ⁡ x ) = arccoth ⁡ ( csc ⁡ x ) . inverse-Gudermannian 𝑥 1 2 𝑥 1 4 𝜋 𝑥 𝑥 hyperbolic-inverse-sine 𝑥 hyperbolic-inverse-cosecant 𝑥 hyperbolic-inverse-cosine 𝑥 hyperbolic-inverse-secant 𝑥 hyperbolic-inverse-tangent 𝑥 hyperbolic-inverse-cotangent 𝑥 {\displaystyle{\displaystyle{\operatorname{gd}^{-1}}\left(x\right)=\ln\tan% \left(\tfrac{1}{2}x+\tfrac{1}{4}\pi\right)=\ln\left(\sec x+\tan x\right)=% \operatorname{arcsinh}\left(\tan x\right)=\operatorname{arccsch}\left(\cot x% \right)=\operatorname{arccosh}\left(\sec x\right)=\operatorname{arcsech}\left(% \cos x\right)=\operatorname{arctanh}\left(\sin x\right)=\operatorname{arccoth}% \left(\csc x\right).}}
    0 references
    DLMF id
    DLMF:4.23.E42
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2apdec
    0 references
    cosecant function
    Defining formula
    csc ⁡ z 𝑧 {\displaystyle{\displaystyle\csc\NVar{z}}}
    xml-id
    C4.S14.E5.m2adec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
    0 references
    cotangent function
    Defining formula
    cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}
    xml-id
    C4.S14.E7.m2adec
    0 references
    inverse hyperbolic cosecant function
    Defining formula
    arccsch ⁡ z hyperbolic-inverse-cosecant 𝑧 {\displaystyle{\displaystyle\operatorname{arccsch}\NVar{z}}}
    xml-id
    C4.S37.E7.m2adec
    0 references
    Q11246
    Defining formula
    arccosh ⁡ z hyperbolic-inverse-cosine 𝑧 {\displaystyle{\displaystyle\operatorname{arccosh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m9adec
    0 references
    inverse hyperbolic cotangent function
    Defining formula
    arccoth ⁡ z hyperbolic-inverse-cotangent 𝑧 {\displaystyle{\displaystyle\operatorname{arccoth}\NVar{z}}}
    xml-id
    C4.S37.E9.m2adec
    0 references
    inverse hyperbolic secant function
    Defining formula
    arcsech ⁡ z hyperbolic-inverse-secant 𝑧 {\displaystyle{\displaystyle\operatorname{arcsech}\NVar{z}}}
    xml-id
    C4.S37.E8.m2adec
    0 references
    Q11247
    Defining formula
    arcsinh ⁡ z hyperbolic-inverse-sine 𝑧 {\displaystyle{\displaystyle\operatorname{arcsinh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m8adec
    0 references
    Q11248
    Defining formula
    arctanh ⁡ z hyperbolic-inverse-tangent 𝑧 {\displaystyle{\displaystyle\operatorname{arctanh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m10adec
    0 references
    inverse Gudermannian function
    Defining formula
    gd - 1 ⁡ ( x ) inverse-Gudermannian 𝑥 {\displaystyle{\displaystyle{\operatorname{gd}^{-1}}\left(\NVar{x}\right)}}
    xml-id
    C4.S23.E41.m2aadec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aldec
    0 references
    secant function
    Defining formula
    sec ⁡ z 𝑧 {\displaystyle{\displaystyle\sec\NVar{z}}}
    xml-id
    C4.S14.E6.m2aadec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    tangent function
    Defining formula
    tan ⁡ z 𝑧 {\displaystyle{\displaystyle\tan\NVar{z}}}
    xml-id
    C4.S14.E4.m2aadec
    0 references
    Q11218
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C4.S1.XMD6.m1mdec
    0 references
     
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