Gudermannian function

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

The Gudermannian function, named after Christoph Gudermann (1798 - 1852), relates the circular and hyperbolic trigonometric functions without resorting to complex numbers. It is defined by

[(x)\,]	[=\int_0^x \frac]
	[=2\arctan \left(\tanh\frac\right)]
	[=2\arctan e^x-.]

Note that

[\tanh\frac = \tan \frac(x)}.\,]

The following identities also hold:

[\sinh(x)=\tan(\mbox(x))\ ]

[\cosh(x)=\sec(\mbox(x))\ ]

[\tanh(x)=\sin(\mbox(x))\ ]

[\mbox(x)=\cos(\mbox(x))\ ]

[\mbox(x)=\cot(\mbox(x))\ ]

[\coth(x)=\csc(\mbox(x))\ ]

The inverse Gudermannian function is given by

[\operatorname(x)]	[=^(x)=\int_0^x \frac\,]
	[=\operatorname(\sec x)=\operatorname(\sin x)\,]
	[=\ln\left(\sec(x)(1+\sin(x))\right)\,]
	[=\ln(\tan x+\sec x)=\ln\left(\tan\left(\frac+\frac\right)\right)\,]
	[=\frac\ln\left(\frac \right).\,]

The derivatives of the Gudermannian and its inverse are

[\,\mbox(x)=\mbox(x),]

[\,\operatorname(x)=\sec(x).]

See also

• hyperbolic secant distribution
• Mercator projection
• tangent half-angle formula
• tractrix
• trigonometric identity

References

• CRC Handbook of Mathematical Sciences 5th ed. pp 323-5.
• , [Gudermannian Function] at MathWorld.

   

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