# Questions tagged [transcendental-functions]

Transcendental functions are those functions that do not satisfy an algebraic equation.

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### Converting an integral to hypergeometric function [closed]

I have encountered an integral as follows $$\int_{0}^{1}{\left(k^{2}x^{2}-k^{2}x+m_{2}^{2}+m_{1}^{2}x-m_{2}^{2}x \right)^{\frac{d-4}{2}}}dx$$ Any suggestion how to convert it into a hypergeometric ...
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### Is this subclass of transcendental function studied in the literature?

By definition, an analytic function $f(x)$ is an algebraic function if there exists a rational non-zero polynomial $P(x,f(x))$ such that $P(x,f(x)) = 0$. If $f(x)$ is not transcendental, then $f(x)$ ...
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### Asymptotics of $\int_{0}^{1} \frac{\tan^{-1}(x^n)}{\sqrt{1 - x^n}} \, dx$

Working around this question, I tried to compute $$I_n=\int_{0}^{1} \frac{\tan^{-1}(x^n)}{\sqrt{1 - x^n}} \, dx$$ which I have been unable to express using Mathematica for general $n$. Some tedious ...
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### Images of a vector under the Galois differential group span the solution set

I am reading the paper "A refined version of the Siegel-Shidlovskii theorem" by F. Beukers. In the proof of Theorem 1.5, he mentions the following results in Galois differential theory. Let ...
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### "Nice" values of x such as sin(x) is transcendental

First time posting here so please forgive any lack of adherence to best practices. sin(x) is a transcendental function. However most common values for the angle x will yield an algebraic number result....
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### Transcendental terms [closed]

What does it mean for two curves parametrised about 0 (say the graph of two functions of real variable) to 'differ only by transcendentally small terms'? How does this relates to their Taylor ...
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### Why can't $a=\theta-\sin(\theta)$ be solved for $\theta$ in terms of $a$ in closed form?

Context: I had taken an interest in alchemical symbols. Many of the ancient drawings are understandably crude, given the tools available at the time. In spite of their rough appearance, I imagined ...
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### How to prove a transcendental equation is never zero or sometimes zero?

I am an engineer by profession. Recently, while working on beams, I obtained the following transcendental equation: $$1-\cos(x)\cosh(x)+x^2\sin(x)\sinh(x)$$. I was wondering if this expression can ...
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### Approximate solutions to a transcendental equation of two variables

Say I have an equation of the form: $$y = A + B\sin (x) +C\sin (x+Dy)$$ on the domain $0<x<2\pi$. I want to get $y$ as a function solely of $x$, i.e. remove the $y$-dependence from the R.H.S. ...
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### Solve transcendental equation: At $α\ll 1$, $\left|\cos x + \alpha \frac{\sin x}{x}\right| > 1$, determine the width of of the $k$-th zone at $k\gg1$.

To solve this transcendental equations approximately: At $\alpha \ll 1$ find the positive solution of inequality: $\left|\cos x + \alpha \frac{\sin x}{x}\right| > 1$, they are divided into series ...
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### Taking the inverse of a one-to-one polynomial

I'm trying to take the inverse of: $$f(x)=\frac{4x^3}{x^2+1}$$ When looking at the graph, it seems to be fully inversible (it is one-to-one), so I should be able to end up with another equation that ...
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### Prove that $e^{\frac{1}{\log(x)}}$ is at least countably transcendental

Q: Prove that $f(x)=e^{\frac{1}{\log(x)}}$ is at least countably transcendental for $x\in\Bbb R\cap (0,1).$
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### Closed-form solvability of elementary transcendental equations?

Fern-Ching Lin ([Lin 1983]) and Timothy Chow ([Chow 1999]) asked, when the solutions of a transcendental equation of elementary functions can be elementary numbers. My question is: To which more ...
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### How can you prove algebraic inputs of a transcendental function are transcendental?

In general, it's not true that a given transcendental function $f(z)$ will give you a transcendental output for countably infinite algebraic inputs, but is there a condition that can prove that ...
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### What is transcendental equation/function?

I looked up several sources on the internet. A transcendental equation is an equation containing a transcendental function of the variable(s) being solved for. Such equations often do not have ...
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### Does this transcendental equation have solutions for a non real variable?

Having solved the transcendental equation $e^{\frac{1}{\log(x)}}=x$ I found that it has solutions for a real variable $x.$ Does it have solutions for not real $x$ (i.e. over the complexes, ...
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