Questions tagged [transcendental-equations]

Transcendental equations are equations containing transcendental functions, i.e. functions which are not algebraic. An algebraic function is a function that satisfies a polynomial equation whose terms are themselves polynomials with rational coefficients.

Transcendental equations are equations containing transcendental functions, i.e. functions which are not algebraic. An algebraic function is a function that satisfies a polynomial equation whose terms are themselves polynomials with rational coefficients.

386 questions
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What is the solution of $\cos(x)=x$?

There is an unique solution with $x$ being approximately $0.739085$. But is there also a closed form solution?
corto
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Trigonometric/polynomial equations and the algebraic nature of trig functions

Prove or disprove that an equation involving one trig function (either $\sin,\cos,\tan$, etc) with an argument of the form $ax+b$ for non-zero rational $a,b$ and a polynomial with non-zero rational coefficients and a constant term not equal to…
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The positive root of the transcendental equation $\ln x-\sqrt{x-1}+1=0$

I numerically solved the transcendental equation $$\ln x-\sqrt{x-1}+1=0$$ and obtained an approximate value of its positive real root $$x \approx 14.498719188878466465738532142574796767250306535...$$ I wonder if it is possible to express the exact…
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Is the positive root of the equation $x^{x^x}=2$, $x=1.47668433...$ a transcendental number?

I can prove using the Gelfond–Schneider theorem that the positive root of the equation $x^{x^x}=2$, $x=1.47668433...$ is an irrational number. Is it possible to prove it is transcendental?
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Find a number $x$ such that $\sum_{n=1}^\infty\frac{n^x}{2^n n!} = \frac{1539}{64}e^{1/2}$

I need to find a number $x$ such that $$\sum_{n=1}^\infty\frac{n^x}{2^n n!} = \frac{1539}{64}e^{1/2}.$$ What is the best approach to this problem?
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A curious equation containing an integral $\int_0^{\pi/4}\arctan\left(\tan^x\theta\right)d\theta=\frac{\ln2\cdot\ln x}{16}$

I came across an interesting problem that I do not know how to solve: Find $x>0$ such that $$\int_0^{\pi/4}\arctan\left(\tan^x\theta\right)d\theta=\frac{\ln2\cdot\ln x}{16}.$$ Could you suggest how to approach it?
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How to solve $\upsilon^\upsilon=\upsilon+1$

What is the real positive $\upsilon$ that satisfies $\upsilon^\upsilon=\upsilon+1$? I think the Lambert-W function might be relevant here, but I have no idea how to use it. $\upsilon\approx 1.775678$ I just really like the letter upsilon. It doesn't…
B H
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Is it possible to prove the positive root of the equation ${^4}x=2$, $x=1.4466014324...$ is irrational?

(somewhat related to my earlier question) Let ${^n}a$ denote tetration $\underbrace{a^{a^{.^{.^{.^a}}}}}_{n \text{ times}}$ (or, defined recursively, ${^1}a=a$, ${^{n+1}}a=a^{({^n}a)}$). The equation ${^4}x=2$ has a positive root…
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Differentiating both sides of a non-differential equation

I'm working on solving for $t$ in the expression $$\ln t=3\left(1-\frac{1}{t}\right)$$ and although I can easily tell by inspection and by graphing that $t=1$, I'd like to prove it more rigorously. I got stuck trying to solve this algebraically, so…
wchargin
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Solving base e equation $e^x - e^{-x} = 0$

So I ran into some confusion while doing this problem, and I won't bore you with the details, but it comes down to trying to solve $e^x - e^{-x} = 0$. I know to solve it, we can rewrite it as $e^x - \frac{1}{e^x} = 0$ and then get LCD so form…
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How do I solve this exponential equation? $5^{x}-4^{x}=3^{x}-2^{x}$

How do I solve this exponential equation? $$5^{x}-4^{x}=3^{x}-2^{x}$$
Young
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How to solve $x!=5^x$?

Or, more generally, $$\Gamma (x+1)=\int_0^{\infty}t^{x}e^{-t}dt=p^x$$ with $p \in \mathbb{Z}^+$ and $x \in \mathbb{C}$. Perhaps begin with $\large p^x=p^x \lim_{n \rightarrow…
Meow
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Global Minimum of $f(a) = \int _{-\infty}^{\infty} \exp\left(-|x|^a\right)dx, a\in(0,\infty)$

Playing around with the Standard Normal distribution, $\exp\left(-x^2\right)$, I was wondering about generalizing the distribution by parameterizing the $2$ to a variable $a$. After graphing the distribution for different values of $a$, I decided to…
David Etler
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Is there a closed form of the Laplace Limit Constant: $x$ such that $\frac{xe^{\sqrt{x^2+1}}}{\sqrt{x^2+1}+1}=1$ using library functions?

The Laplace Limit Constant $\lambda$ is well know constant which is the $y$ value of the global extrema of: $$x\,\text{sech}(x):$$ Therefore: $$x=\max(x\,\text{sech}(x))=-\min(x\,\text{sech}(x))=1.19967…\implies y= 1.19967… \text{sech}(1.19967……
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Formal proof that $e^x$ is not algebraic

How do I give a formal proof that $e^x$ is not algebraic, like for example: $$\sum_{n\geq0}\frac{x^n}{n!}\notin\mathbb{C}_{\mathrm{alg}}[[x]]$$ Help appreciated!
UFO Hunter
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