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$e^{-\beta \frac{1}{2}\hbar \omega} \frac{1}{1 - e^{\beta \hbar \omega}} = \frac{1}{2 \sinh \left( \frac{\beta \hbar \omega}{2}\right)}$

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#trigonometry #hyperbolic-functions
113
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Comparing Coefficients to find an exact solution

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#asymptotics #factoring #hyperbolic-functions #perturbation-theory
138
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Calculation of capacitance between two cylinders

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#trigonometry #systems-of-equations #physics #hyperbolic-functions #electromagnetism
1.5k
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The 'c' in the spelling of $\operatorname{arsinh}(x)$ vs $\operatorname{arcsinh}(x)$.

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#notation #hyperbolic-functions
59
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Is $\frac{1}{1+0.1x}$ hyperbolic in $x$?

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#functions #trigonometry #hyperbolic-functions
16k
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Relationship between $\sin(x)$ and $\sinh(x)$

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#hyperbolic-functions
126
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Hyperbolic functions proof using $\cosh x$

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#hyperbolic-functions
106
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Proof Involving Hyperbolic Functions

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#polynomials #hypergeometric-function #hyperbolic-functions
637
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Fourier Transform of $ e^ {t/2} \operatorname{sech}(t) $

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#fourier-analysis #fourier-transform #hyperbolic-functions
179
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$\mathrm{arctanh} (x+y) = ?$

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#inverse-function #hyperbolic-functions
83
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Hyperbolic Trig Geometry

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#geometry #definition #graphing-functions #hyperbolic-geometry #hyperbolic-functions
103
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Finding circles with hyperbolic functions and hyperbolas with trig functions

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#trigonometry #hyperbolic-functions
50
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Equation involving hyperbolic tangents

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#trigonometry #closed-form #hyperbolic-functions
177
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How to solve this complex nonlinear system of equations? (comes from solving catenary equations)

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#systems-of-equations #hyperbolic-functions
179
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Compare the heights of two midpoints in a hyperbolic triangle

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#trigonometry #triangles #hyperbolic-geometry #hyperbolic-functions

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