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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Why a transcendental equation can not be analytically evaluated

I'm reading this book in Classical Mechanics and they derive an equation for the time a projectile takes to reach the ground once is fired (accounting for air resistance): ...
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Can we solve $ae^x+b=x^{-\alpha} $ using LambertW function?

In the equation, a,b,$\alpha$ are all constants. Can we solve the equation?
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27 views

It is possible to talk about the degree of a transcendental equation?

When we deal with algebraic equations involving polynomial and so on we know what the degree of the equation is and this tells us how many solutions we'll find (at least in complex numbers). But this ...
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40 views

Solving Kepler's Equation

I've been working on simulating orbits. I've found that, when solving Kepler's equation, $M = E - \varepsilon\sin{E}$, I'm unsure about the solution to use. For a true anomaly $< \pi$, using the ...
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6answers
239 views

How to solve $4^x+\sin(x)=10$

$$4^x+\sin(x)=10$$ I would use a log function to solve it but I don't know what to do with $\sin(x)$. What is the $x$ value of the exponent?
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2answers
45 views

Solving $x^y = y^x$ analytically in terms of the Lambert $W$ function

I'm interested in deriving the solution for $y$ in terms of $x$ given $x^y = y^x$ using the Lambert $W$ function. Wolfram Alpha states: $$y = - \frac{x\ W\left(-\frac{\log(x)}{x}\right)}{\log(x)}$$ ...
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1answer
55 views

the roots & the limit of $2^{x^{\cos(x)}}\sqrt{\cos(x)}=2^{x}$

If $$2^{(2\pi)^{\cos(2\pi)}}\sqrt{\cos(2\pi)}=2^{2\pi}$$ Can you obtain or is it plausible to find the roots and the limit of $$2^{x^{\cos(x)}}\sqrt{\cos(x)}=2^{x}$$ if $0 < \cos(x)$ and $0 < ...
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1answer
39 views

How to calculate the shielding time and determine the time step

The problem is illustrated as follows. A shielding plate scans over a target plate at a constant speed $v_{scan}$ and dynamically shadows the target plate to adjust the exposure time of the light ...
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1answer
76 views

Solving an equation including $e^{-x}$ with the Lambert W function

Given two functions of $x$, namely $f(x)$ and $g(x)$, where $$f(x)=x^2-4x+8$$$$g(x)=3xe^{-x}$$ the shortest distance between the graphs of the functions is sought. I begin by defining a function ...
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3answers
167 views

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 ...
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159 views

Solving $\ln{x}=\tan{x}$ with infinitely many solutions

Lets take $f(x)=\ln{x}$ and $g(x)=\tan{x}$ When $f(x)=g(x)$ that is $\ln{x}=\tan{x}$, we see that the graph is like: Hence we see that there are infinitely many solutions to $x$ but the two ...
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56 views

what is the best approximation for sine?

can you tell me which is the best approximation for cosine/sine functions. It should also reduce the computational complexity. I've already tried the Bhaskara-1 approximation. Can you suggest me ...
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3answers
129 views

How to solve $x^2 = e^x$

The question is to find $x$ in: \begin{equation*} x^2=e^x \end{equation*} I know Newton's method and hence could find the approx as $x\approx -0.7034674225$ from \begin{equation*} ...
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2answers
239 views

Solution of a Lambert W function

The question was : (find x) $6x=e^{2x}$ I knew Lambert W function and hence: $\Rightarrow 1=\dfrac{6x}{e^{2x}}$ $\Rightarrow \dfrac{1}{6}=xe^{-2x}$ $\Rightarrow \dfrac{-1}{3}=-2xe^{-2x}$ ...
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4answers
68 views

Always transcendental? $a + b e^{-x} - f(x) = 0$

I want to choose $f(x)$ such that the equation $$a + b e^{-x} - f(x) = 0$$ is analytically solvable. Ideally, I want $f(x)$ to be some function that is symmetric about 0 and everywhere positive, like ...
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3answers
71 views

Ugly expression. Cant tell if I can simplify. Is there a general method for simplifying basic algebraic expressions?

Let $\rho, \omega, c > 0$ and let $\alpha \in [0,1]$. I have managed to calculate \begin{align*} \frac{\rho}{\alpha x^{\alpha-1}y^{1-\alpha}}&= \eta\\ \frac{\omega}{(1-\alpha) ...
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0answers
27 views

Proof that for $\alpha > 0$ and $\tau \geq 0$ , P($s$) = $(s+\alpha+ e^{-s\tau})$ has no solution less than zero

P($s$) = $(s+\alpha+ e^{-s\tau})$ Proof that for $\alpha > 0$ and $\tau \geq 0$ the polynomial has no solution less than zero. I am having difficulty proving that for $\alpha > 0$ and $\tau ...
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3answers
284 views

Request for help to solve an equation with LambertW: $ (x^2-4\,x+6) e^x =y$

I want to solve the following equation: $$ (x^2-4\,x+6) e^x =y \tag{1} $$ It looks a bit like the following equation: $$ x e^x =y \tag{2} $$ Since the solution of equation (2) is: x=LambertW(y), ...
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1answer
110 views

Can I solve this with a Lambert Function?

New to W-Functions and do not understand it properly. How do I solve this equation? I know about numerical solutions (or graph solution), but I'm interested in pure algebraic solution if it exists. ...
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3answers
72 views

For what k values are there more than one solution to the following equation?

Suppose we have the equation, where k is some constant: $0 = (x+k)e^{-(x+k)^2} + (x-k)e^{-(x-k)^2} + xe^{-x^2}$ A trivial solution exists, where x = 0. How can I figure out what range of values for ...
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2answers
110 views

How to solve this equation for x? $0 = (x+k)e^{-(x+k)^2}+(x-k)e^{-(x-k)^2}$

How can I solve this equation: $0 = (x+k)e^{-(x+k)^2}+(x-k)e^{-(x-k)^2}$ To find x in terms of k?
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2answers
55 views

Solving $z=w/2-\sin(tw)/(2t)$ for $w$

Is it possible to solve $$z=\frac{w}{2}-\frac{\sin(tw)}{2t},$$ for $w$? My first thoughts were that we would have to be careful about the domain of $f(w)$ so that the inverse was actually a function ...
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1answer
77 views

How to solve the following equation? $\left(\sqrt{u^2-1}+u\right)^{1/u}=\pi ^{1/\pi }$

I have no clue: $$\left(\sqrt{u^2-1}+u\right)^{1/u}=\pi ^{1/\pi }$$
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2answers
53 views

Equations involving factorial/Gamma function

Are there any known methods to formally solve equations like: 1)$x^3!+(2x^2)!-x!+3=0$ 2)$x!=e^x$ ($0$ is trivial but there must be another one) 3)$(2x!)^2+x!-1=0$ 4)$x!!+x!=7$ I don't need ...
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39 views

Understanding simple transcendental field extensions [duplicate]

I would like to understand the definition of a simple transcendental extension and the theorem that states all such extensions are isomorphic. So for example, if $K \subseteq \mathbb{C} $ is any ...
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1answer
72 views

find an inverse function of complicated one

Let $f:\mathbb{R}\rightarrow \mathbb{R}$: $$f(x) = \sin (\sin (x)) +2x$$ How to calculate the inverse of this function? So far i searched a lot in the internet but i didn't find any easy algorithm ...
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1answer
65 views

Two kind of equations involving natural log and exponentiation

I know how to solve equations using Lambert's W function like $xe^x=k$ or $e^x+x=k$ But how can I solve this two kinds of equations involving natural log ? $e^x \ln(x)=k$ and $e^x+\ln(x)=k$ I ...
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40 views

Solving the transcendental equation $Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$.

I need to solve the following equation: $Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$ for $x\in\mathbb{R}^{\ast}$ and where $k\in\mathbb{R}^{+}$. Here $Li_{3}$ and $Li_{2}$ are the third and ...
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1answer
80 views

Transcendental equations involving more than 2 terms

I now how to solve transcendental equations involving only two terms like: $xe^x=k$ $x=W(k)$ Where W(x) is the Lambert's Omega function. But how can I solve (for $x$) a more general case? Like: ...
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2answers
33 views

Solving equation for x values

What methods I can use to solve for $x$? And how to do it? $$2x + ( 1 + \cot(x/2) ) / \sin(x/2) + 0.8 = 0$$
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3answers
129 views

explicit solution for transcendental equation

Does anyone knows whether there is an explicit, analytical solution for transcendental equations of the form $A x + B \tanh(C x) + \coth(x) = 0$, where $A, B$, and $C$ are positive real constants?
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2answers
78 views

Solution to $x^\alpha + p x = q$?

I was wondering if there was any tricks, similar in spirit to the Vieta's substitution, that would apply the equation $$ x^\alpha + p x = q, $$ where $p,q$ and $\alpha$ are real constants. In ...
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3answers
169 views

Exact values of the equation $\ln (x+1)=\frac{x}{4-x}$

I'm asking for a closed form (an exact value) of the equation solved for $x$ $$\ln (x+1)=\frac{x}{4-x}$$ $0$ is trivial but there is another solution (approximately 2,2...). I've tried with ...
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26 views

solving/approximating the transcendental inequality $c \le αx + β(b^x) + γx(b^x)$

I couldn't find a representation of $x$ using Lambert $W$ function and I doubt this is even possible. Assuming there is no clean solution and numerical methods must be used, is there a way to ...
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56 views

Solving transcendental equation with the unknown on both sides

I need to find the point of intersection between a line segment and a sine wave. Line: $y=-2x+1$ Wave: $y=\sin x$ I put both equations together. $-2x+1=\sin x$ I attempt to isolate $x$. $\sin ...
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39 views

Determine if the graph $f(x) = \ln(x)$ has any critical numbers

Determine if the graph $f(x) = \ln(x)$ has any critical numbers. The derivative would be $f'(x) = \frac{1}{x}$
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258 views

How to solve: $\frac{x}{\log_2(x) }= y$

For example, I can solve: $x \log_2(x) = y$ $x \log_2(x) = x \log_e(x) / \log_e(2) = e^{\log_e(x)} \log_e(x) / \log_e(2)$ $e^{\log_e(x)} \log_e(x) = y\log_e(2)$ $e^{W(z)} W(z) = z$, where W(z) is ...
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2answers
53 views

Find roots for an equation with quadratic, linear and log terms?

I'm wondering if there exists a closed-form or analytic expression for the roots of an equation of the form $ax^2 + bx + c\log x=0.$ considering the natural $\log$. Wolfram alpha is leading me to ...
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2answers
145 views

Solving the equation $\ln(x)=-x$

I tried solving this equation for a long time but did not succeed. Any help is appreciated. $$\ln x=-x$$ I am not sure the tag is correct, I am not familiar with English mathematical terms. Please ...
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1answer
59 views

An equation with multiple solutions: finding the maximum of the function of the solutions.

Possibly, this is a bad (stupid) question, but sometimes some discussion helps. I have a fixed point equation (involving $\tanh$). I would like to derive the dependency of some function of the fixed ...
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60 views

Why a trigonometric function doesn't satisfy a polynomial equation?

Why can't I have a trigonometric function as an input to a polynomial equation?
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47 views

Showing no algebraic solution exists for a given equation

Let $f(x)=g(x)$ be an equation (1) where at least one of $f$ and $g$ are transcendental functions. Let $h(x)=f(x)-g(x)$. If it can be shown that $h^{-1}(0)$ is non-algebraic, that implies that there ...
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1answer
76 views

Solving equation involving self-exponentiation

How do I solve the equation $\displaystyle x=ay^2(by)^{\frac 1y}$ for $y$, where $a$ and $b$ are constants? I've been trying to manipulate this into a form on which I can use the Lambert W function, ...
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51 views

Specialized numerical method for transcendental equation

Is there any specialized, very fast, numerical method for solving equations of a type $$ e^{-px-q} = \frac{ax^2 + bx + c}{kx + l} $$ wher all $ a, p, q $ are strictly positive? To be more precise, ...
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1answer
32 views

Characterizing conditions for $\tanh{(kx-b)}=x$ to have 1/2/3 fixed points.

I am trying to understand what are the conditions for $\tanh{(kx-b)}$ to have 1 or 2 or 3 fixed points. That is I am trying to characterize conditions on $k$ and $b$ for which equation ...
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147 views

How to solve Kepler's equation $M=E-\varepsilon \sin E$ for $E$?

I'm trying to create a program to solve a set of Kepler's Equation and I cannot isolate the single variable to use the expression in my program. The Kepler Equation is $$M = E - \varepsilon ...
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3answers
69 views

A formal way to solve a transcendental equation

Is there a formal way of solving the equation $$x^x = \frac{1}{\sqrt{2}}\ ?$$The solutions are $x = \frac{1}{4},\frac{1}{2}$. This can be easily obtained by plotting the function or just by guessing ...
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1answer
138 views

Inverse of $f(x) = xe^x-x$

I'm wondering if there is a way to obtain the inverse of the function $y=xe^x-x$. I am aware of the use of Lambert's W function in the inverse of $xe^x$ but as can be seen this is a different animal ...
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2answers
47 views

Solving equation $-t-0.2+ 0.2e^t=1$

I don't know how to solve this one, please give me some clues. $-t-0.2+ 0.2e^t=1$ $e^{-t}(0.16e^{2t}+0.48e^t+0.36)=1$
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2answers
100 views

Solving a transcendental equation

How do I go about solving the following equation? $$x = A + B \log\left( \cosh\left(\frac{x}{C}\right)\right)$$