Inverse Gudermannian

The inverse Gudermannian $\mathrm{gd}^{-1}$ is the inverse function of the Gudermannian function. It may be defined by the following formula for $x \in \mathbb{R}$: $$\mathrm{gd}^{-1}(x)=\displaystyle\int_0^x \dfrac{1}{\cosh(t)} dt,$$ where $\cosh$ denotes the hyperbolic cosine.

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  Graph of $\mathrm{gd}^{-1}$.

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  Domain coloring of $\mathrm{gd}^{-1}$.

Properties

Theorem

The following formula holds: $$\sinh(\mathrm{gd}^{-1}(x))=\tan(x),$$ where $\sinh$ is the hyperbolic sine, $\mathrm{gd}^{-1}$ is the inverse Gudermannian, and $\tan$ is the tangent.

References

Theorem: The following formula holds: $$\cosh(\mathrm{gd}^{-1}(x))=\sec(x),$$ where $\cosh$ is the hyperbolic cosine, $\mathrm{gd}^{-1}$ is the inverse Gudermannian, and $\sec$ is the secant.

Proof: █

Theorem: The following formula holds: $$\mathrm{tanh}(\mathrm{gd}^{-1}(x))=\sin(x),$$ where $\mathrm{tanh}$ is the hyperbolic tangent, $\mathrm{gd}^{-1}$ is the inverse Gudermannian, and $\sin$ is the sine.

Proof: █

Theorem: The following formula holds: $$\mathrm{csch}(\mathrm{gd}^{-1}(x))=\cot(x),$$ where $\mathrm{csch}$ is the hyperbolic cosecant, $\mathrm{gd}^{-1}$ is the inverse Gudermannian, and $\cot$ is the cotangent.

Proof: █

Theorem: The following formula holds: $$\mathrm{sech}(\mathrm{gd}^{-1}(x))=\cos(x),$$ where $\mathrm{sech}$ is the hyperbolic secant, $\mathrm{gd}^{-1}$ is the inverse Gudermannian, and $\cos$ is the cosine.

Proof: █

Theorem: The following formula holds: $$\mathrm{coth}(\mathrm{gd}^{-1}(x))=\csc(x),$$ where $\mathrm{coth}$ is the hyperbolic cotangent, $\mathrm{gd}^{-1}$ is the inverse Gudermannian, and $\csc$ is the cosecant.

Proof: █