Gudermannian Function

$\{\backslash rm\; gd\}(x)=\backslash int\_0^x\; \backslash frac\{dt\}\{\backslash cosh\; t\}$

$\{\}=2\backslash arctan\; \backslash left(\backslash tanh\backslash frac\{x\}\{2\}\backslash right)$

$\{\}=2\backslash arctan\; e^x-\{\backslash pi\backslash over2\}.$

$\backslash tanh\backslash frac\{x\}\{2\}\; =\; \backslash tan\; \backslash frac\{\backslash mbox\{gd\}(x)\}\{2\}.$

$\backslash sinh(x)=\backslash tan(\backslash mbox\{gd\}(x))\backslash$

$\backslash cosh(x)=\backslash sec(\backslash mbox\{gd\}(x))\backslash$

$\backslash tanh(x)=\backslash sin(\backslash mbox\{gd\}(x))\backslash$

$\backslash mbox\{sech\}(x)=\backslash cos(\backslash mbox\{gd\}(x))\backslash$

$\backslash mbox\{csch\}(x)=\backslash cot(\backslash mbox\{gd\}(x))\backslash$

$\backslash coth(x)=\backslash csc(\backslash mbox\{gd\}(x))\backslash$

$\{\backslash rm\; gd\}^\{-1\}(x)=\backslash int\_0^x\; \backslash frac\{dt\}\{\backslash cos\; t\}=\backslash ln(\backslash tan\; x+\backslash sec\; x).\backslash ,$

$\{d\; \backslash over\; dx\}\backslash ,\backslash mbox\{gd\}(x)=\backslash mbox\{sech\}(x)$

$\{d\; \backslash over\; dx\}\backslash ,\backslash mbox\{gd\}^\{-1\}(x)=\backslash sec(x)$

See also

• trigonometric identity
• tangent half-angle formula
• tractrix
• Mercator projection

References

• CRC Handbook of Mathematical Sciences 5th ed. pp 323-5.

External links

• Gudermannian Function at MathWorld

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