Gudermannian function

Gudermannian function with its asymptotes y = ±π/2 marked in gray.

The Gudermannian function relates the circular and hyperbolic trigonometric functions without resorting to complex numbers. It is defined by

${\rm {gd}}(x)=\int _{0}^{x}{\frac {dt}{\cosh t}}$

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

${}=2\arctan e^{x}-{\pi \over 2}.$

Note that

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

The following identities also hold:

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

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

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

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

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

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

The inverse Gudermannian function is given by

${\rm {gd}}^{-1}(x)=\int _{0}^{x}{\frac {dt}{\cos t}}=\ln(\tan x+\sec x).\,$

The derivatives of the Gudermannian and its inverse are

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

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

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