Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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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…
Simply Beautiful Art
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How to solve $a^x+b^x=1$ (solve for $x$)

I am sorry if this is too easy question for this site, but I really can't find the solution... $a^x+b^x=1$ Tyma Gaidash asked for context: I don't have much to add, I was trying to understand how fast population grow assuming that everyone born…
מתן ל
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Are these polynomials already known?

In the frame of a more complicated problem, I need to find the zero of function $$f(x)=\Gamma(k+2-x)-k\,\Gamma(k+1)$$ which is the same as the zero of function $$g(x)=\log \big((\Gamma (k+2-x)\big)-\log \big(k \,\Gamma (k+1)\big)$$ This does not…
Claude Leibovici
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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?
Vladimir Reshetnikov
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Inverse function of $y=\frac{\ln(x+1)}{\ln x}$

I've been wondering for a while if it's possible to find the inverse function of $y=\frac{\ln(x+1)}{\ln x}$ over the reals. This is the same as finding the positive real root of $x^y-x-1$. I realize that it's impossible with elementary functions,…
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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 $x^x = (x-1)^{x+1}$?

Background: I was trying to estimate the size of $21^{21}$ for some problem and decided to use $20^{22}$ as hopefully a rough approximate ($20^{22} = 2^{22} \cdot 10^{22} \approx 10^{28}$). But then I wanted to see if that was over or underestimate…
mpear617
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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…
OlegK
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Is there a way to solve $\sin(x)=x$?

Note: Question was originally to solve it algebraically, though I've decided to change it to analytically due to the comments and answers. When trying to solve $\sin(x)=x$, the obvious first solution is $x=0$. There are, however, an infinite…
Simply Beautiful Art
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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?
Laila Podlesny
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Prove the sum $\sum_{n=1}^{\infty} \left(\frac{\sin x_n-\sinh x_n}{\cos x_n+\cosh x_n}\right)^2\frac{1}{x_n^6}=\frac1{80}$ evoked by equation

An elegant result gives $$\sum_{n=1}^{\infty} \left(\frac{\sin x_n-\sinh x_n}{\cos x_n+\cosh x_n}\right)^2\frac{1}{x_n^6}=\frac1{80}$$ where $x_n>0$ is the real root of equation $$ \cos x \cosh x + 1=0 $$ One obvious observation says this equation…
Nanayajitzuki
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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…
Astro
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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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Solve $2^x=x^2$

I've been asked to solve this and I've tried a few things but I have trouble eliminating $x$. I first tried taking the natural log: $$x\ln \left( 2\right) =2\ln \left( x\right)$$ $$\dfrac {\ln \left( 2\right) }{2}=\dfrac {\ln \left( x\right)…
seeker
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