# Tagged Questions

For questions related to hyperbolic functions: $\sinh$, $\cosh$, $\tanh$, and so on.

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### A improper integral with integrable singularities

Let $\alpha$ and $v_0$ be both positive. Consider a following integral: {\mathcal J}^\alpha_{1/2}(v_0) := \int\limits_0^\infty v^{\alpha-1} e^{-v} \frac{1}{\sqrt{v_0-v}} d v ...
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### How to simplify expressions like $\sinh(4\,\text{arcsinh}(x))$?

I understand that expressions like $\sinh(\text{arcsinh}(x))$ simplify immediately and expressions like $\sinh(\text{arccosh}(x))$ tend to simplify after some algebra. However I cannot work out how ...
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### Suppose $z=re^{iθ}$, prove : $|e^{iz}|=e^{-r\sinθ}$.

Suppose $z=re^{iθ}$, prove $|e^{iz}|=e^{-r\sin θ}$. I tried but the result was not as expected: $$e^{iz}=\cosh z+i(\sinh z)\\|e^{iz}|=(\cosh z)^2+[i(\sinh z)]^2=\cosh(2z)$$
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### Exponential Proof

Let $c(x)=\dfrac{3^x+3^{-x}}{2}$ and $s(x)=\dfrac{3^x-3^{-x}}{2}$. Show that $(c(x))^2=\frac{1}{2}(c(2x)+1)$. How does one go about solving this? I have honesty tried substituting in ...
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### Law of Cosines with imaginary arguments?

Does the law of cosines: $c^2 = a^2 + b^2 - 2 a b \cos \theta$ work with imaginary angles? to get something like: $c^2 = a^2 + b^2 - 2 a b \cosh \theta$ Alternatively, is there a hyperbolic ...
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### Evaluate $\int\frac{1}{x}\coth(ax)\sin^2(\frac{xt}{2})dx$

I am trying to integrate this function: $\int_0^\infty\frac{1}{x}\coth(\frac{\hbar x}{2kT})\sin^2(\frac{xt}{2})dx$ which Wolframalpha (for me) returns nothing, just a blank screen. I thought that it ...
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### Lateral limits of function involving hyperbolic trignometric functions

I am not being able to calculate the lateral limits at 0 of the following function $f(x) = \frac{\sinh(x)}{2\sqrt{\cosh(x) - 1}}$ I have tried both direct substitution (yields 0/0) and L'Hospital's ...
For $y : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ and let $j : \mathbb{R} \to \mathbb{R}$ $$\frac{\partial y}{\partial x_2} + \frac{\partial (j \circ y)}{\partial x_1} = 0$$