## 17Calculus - Gudermannian Function

##### 17Calculus

The Gudermannian Function relates the hyperbolic functions to the trig functions without complex numbers. There are many properties and identities related to this function. Check the related topics and links tab for links.

$$\displaystyle{ gd(x) = \int_{0}^{x}{ \frac{1}{\cosh(t)} ~ dt } }$$

This video discusses the above definition integral (there are several definitions) as well as these integrals. $gd(x) = 2\arctan(e^x) - \pi/2$ $\sin(gd(x)) = \tanh(x)$ $gd^{-1}(x) = \int_{0}^{x}{ \frac{1}{\cos(t)} ~ dt }$

### Michael Penn - the Gudermannian function!

video by Michael Penn

I am amazed that I was never introduced to this function as an undergrad or in grad school! It is fascinating!

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