Difference between revisions of "Gudermannian"
From specialfunctionswiki
(→Properties) |
|||
| Line 1: | Line 1: | ||
The Gudermannian $\mathrm{gd}$ is defined for $x \in \mathbb{R}$ by the formula | The Gudermannian $\mathrm{gd}$ is defined for $x \in \mathbb{R}$ by the formula | ||
| − | $$\mathrm{gd}(x) = \displaystyle\int_0^x \dfrac{1}{\cosh t} \mathrm{d}t$$ | + | $$\mathrm{gd}(x) = \displaystyle\int_0^x \dfrac{1}{\cosh t} \mathrm{d}t,$$ |
| + | where $\mathrm{cosh}$ denotes the [[cosh|hyperbolic cosine]]. | ||
<div align="center"> | <div align="center"> | ||
Revision as of 18:01, 6 June 2016
The Gudermannian $\mathrm{gd}$ is defined for $x \in \mathbb{R}$ by the formula $$\mathrm{gd}(x) = \displaystyle\int_0^x \dfrac{1}{\cosh t} \mathrm{d}t,$$ where $\mathrm{cosh}$ denotes the hyperbolic cosine.
Graph of $\mathrm{gd}$.
Domain coloring of $\mathrm{gd}$.
Properties
Derivative of Gudermannian
Taylor series for Gudermannian
Relationship between sine, Gudermannian, and tanh
Relationship between cosine, Gudermannian, and sech
Relationship between tangent, Gudermannian, and sinh
Relationship between csc, Gudermannian, and coth
Relationship between secant, Gudermannian, and cosh
Relationship between cot, Gudermannian, and csch
