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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How to show that this Jacobian elliptic function always has two real zeroes?

I have the following elliptic function $f(x)$: $$f(x) = -\ dn^{2}x \ cn^{2}x \left(3m \ sn^{2} x + \frac{E'}{K'}\right) + \ sn^{2} x \left(m \ cn^{2} x + \ dn^{2} x \right) \left(m \ sn^{2} x + \frac{...
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24 views

How to simplify the following expression involving Jacobian elliptic functions?

I would like to show that a certain elliptic function $F(x)$ (that is periodic, say with some period $h$) has exactly two zeroes in $[0,h)$. Let us recall some notation. Given a parameter $m \in [0,1]$...
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13 views

Parameterization of transcendental curves

My question relates to the Wikipedia article on Transcendental curves: https://en.wikipedia.org/wiki/Transcendental_curve . Namely the statement: "Here for a curve, C, what matters is the point set (...
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2answers
56 views

Prove that $e$ is the root of the equation $\int_0^{\infty} \frac{dt}{(t+x)\sqrt{4t+(x+1)^2}}=\frac{1}{x-1}$

It seems numerically that $e$ is the only real root of the equation: $$\int_0^{\infty} \frac{dt}{(t+x)\sqrt{4t+(x+1)^2}}=\frac{1}{x-1}$$ Mathematica confirms it at least to the large number of ...
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19 views

solving equation with variable on both sides

Is there a way to solve the following equation for $v_n^2$? I am working through a problem and I feel like there should be a way to solve it, but I am not sure how to do it: $$v_n^2-(1+\eta)=-i\...
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1answer
41 views

Please help me to solve the equation $e^{-\frac{x}{0.026}}=7.477 x^2 + 0.0146 x$ [closed]

This equation has $e^x$,$x^2$ and $x$ terms.So how can we solve this type of equation?
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2answers
36 views

What is the solution of $\sin z=\cosh 4$?

What is the solution of $\sin z=\cosh 4$? By putting $z=x+iy$ I managed to find that the real part of $z$ is $x= \frac \pi 2+2n\pi $, but the imaginary part is contradictory giving negative value of $...
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4answers
231 views

A String Tied Around The Earth

Say you're standing on the equator and you have a string below you tied around the equator (40,075 km) that is the length of the equator + 1 meter (40,075.001 km). What is the maximum height you can ...
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0answers
22 views

Solving the finite square well graphically (Transcendental Equations)

I am attempting to comprehend how they are solving these transcendental equations here. The equations they derive are $\tan(ka)=\frac{\alpha}{k}$ which gives the even solutions and $-\mbox{cotan}(ka)=\...
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1answer
29 views

Area between $ y=\sin \left(\frac{\pi x}{2}\right)$ and $y=x^2$, functional equations

Find the Area between $ y=\sin \left(\frac{\pi x}{2}\right)$ and $y=x^2$ So I know how to solve this specific case, but one thing I'm not sure of let's say we have: $y=x^2$ and $\:y=\sin \:\left(ax\...
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1answer
34 views

Asymptotic expansion of roots of function

Find expansions for all roots of the equations below as epsilon → 0 with two nonzero terms in each expansion I don't see how drawing the graph will help. Also how do I go about balancing the sizes ...
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1answer
76 views

Solve $\cos 2x = \frac{2x}{3}$ [closed]

I'm trying to solve for $x$ in the following equation: $$\cos 2x = \frac{2x}{3}$$ Any help would be much appreciated!
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0answers
48 views

Is this trigonometric expression always strictly positive?

Let us define a function $f(k,n)$ by \begin{equation} f(k,n)=n \left (\cos\frac{k\pi}{n}\right) \left(1-\cos\frac{k\pi}{n}\right) - \sin \frac{k\pi}{n} \end{equation} Where $\frac{k}{n}$ is ...
2
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3answers
54 views

Number of real roots of the equation $2\cos(x) =(2^x+2^{-x})/2$ [closed]

What's the number of real roots of the equation $2\cos(x) =(2^x+2^{-x})/2$? If any questions ask about real root what is the main thing to check first, and most important? Any particular way to ...
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4answers
148 views

Are there real solutions to $x^y = y^x = 3$ where $y \neq x$?

I need to solve the following equation for (x,y) $$x^y = y^x = 3$$ Everytime I run a numerical method for this problem, I get $$ (x,y) = (1.82546...,1.82546..) $$ I expect there to be a solution ...
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3answers
114 views

Solving $x \ln x=25$ [closed]

Can someone help me solve the following equation? $$x \ln x = 25$$
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1answer
35 views

Finding the number of roots of the equation [duplicate]

The equation $x^{13}-e^{-x}+x-\sin{x}=0$ has No real root More than two real roots. Exactly two real roots. Exactly one real root. I tried doing with the odd derivative and check whether the ...
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0answers
27 views

A problem on Newton-Raphson method

The function $f(x)=0$ has a simple root in the interval $(1,2)$. The function $f(x)$ is such that $|f(x)|>3$ and $|f''(x)|\leq 4$ for all $x\in (1,2)$. Assuming that the Newton-Raphson method ...
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1answer
45 views

How to find the roots of a equation involving log terms?

This question was in my test and I am not sure what to do with it let $f:(0,\infty)\rightarrow \mathbb{R}$ be given by $$f(x)=\log x-x+2$$ then its number of roots of $f$are. So putting it ...
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1answer
140 views

Exact solution to system of first order ODEs

I am trying to solve a system of 1st order ODE's. $m(r)$ and $P(r)$ are real functions of $r$ and should be positive. $a$ and $b$ are just constants. $$\frac{d m(r)}{dr}=\pi(\frac{P(r)}{a}+b)r^2\\$$ ...
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1answer
63 views

How to know if I can't solve an equation with “standard” methods?

I'm particularly fascinated by transcendental equations whose posses closed form solutions and when I pose some of them to my friends or teachers I heard a lot of "You can't solve this in closed form" ...
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110 views

The roots of equation $x3^x=1$ are

I have to find roots of equation $x3^x=1$ A.Infinitely many roots B.$2$ roots C.$1$ root D. No roots\ How do i start? Thanks
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42 views

Self-consistency transcendental equation for Curie-Weiss model

In physics, the ferromagnetic Curie-Weiss spin model leads to a transcendental self-consistency equation for the magnetization $m$ of the form $$ m = \tanh(J m +h)\ , $$ with $J>0$ and $h\in\mathbb{...
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2answers
37 views

To find roots of given equation

If $f(x)=x^2$ and $g(x)=x\sin(x)+\cos(x)$ then i have to find number of points$(x)$ such that $f(x)=g(x)$ I write $h(x)=f(x)-g(x)$. So $h(x)=x^2-x\sin(x)-\cos(x)$ $h'(x)=x(2-\cos(x))$. Since $2-\cos ...
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2answers
52 views

To find number of real solutions of equation $\left(\frac {9}{10}\right)^x =-3+x-x^2 $

To find real solutions of $\left(\frac{9}{10}\right)^x = -3 + x -x^2$ I differentiate it to get $\left(\frac {9}{10}\right)^x log(\frac{9}{10})-1 + 2x=0 $ As x goes to $+\infty$ this goes to $+\...
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0answers
56 views

Solving for a $v$ in $\sum a_i e^{b_i (z^2+d_i) + c_i v}$

I have an equation in complex domain, $$P(e^u,e^v)=\sum_{i=1}^{N} a_i e^{b_i u + c_i v}=0 \;\;\;\text{(A)}$$ and by redefining, at the roots (I'm only showing work for one root), the first ...
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68 views

Genus of trancendental curve

I understand that an algebraic curve is genus 0 iff it can be parameterized using rational functions. I am curious if there is a simple way to know if a transcendental curve is genus 0? Specifically ...
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2answers
68 views

For what values of $x$ is $\cos x$ transcendental?

For what values of $x$ is $\cos x$ transcendental? Is there any way I can figure out the values of $x$ where $\cos x$ is transcendental or do I have to check individually for every $x$ whether it is ...
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1answer
53 views

Difficulty in solving transcendental equation

Let $A,B,C$, and $D$ be positive constants. What's the most concise way to express $x$ in the equation below? $$ A = B\arctan(x/C)+Dx,$$ where $0<x<1$ and we know that $C=\cos(30^\circ)$.
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1answer
69 views

Find the positive root of the equation $\cosh x+\cos x-3=0$, other than numerically

I know you are able to find the root of the equation by using Newton-Raphson method. But is there any other way? $$\cosh x+\cos x-3=0$$ I thought maybe you could say that $-1\leq \cos x \leq 1$. So ...
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0answers
38 views

Existence of solution for equation with erfc

I'm looking for the self-consistent (e.g. input needs to be the same as output) solution of $r$ in $$ r = \frac{1}{2}\operatorname{erfc}(z(r)) $$ where $\operatorname{erfc}$ is complimentary error ...
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1answer
97 views

Table Maker's Dilemma

I am trying to understand the TMD based on this doc. As per my understanding, For transcendental functions i..e (log2 log10 1oge, exp,sin, cos,tan...) the exact value for an input cannot be computed....
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118 views

Derivative of Inverse Mills Ratio (Conditional expectation of normal distrbution is strictly increasing)

I'm trying to show that the derivative of the inverse Mills Ratio is bounded between zero and one. Essentially, for a standard normal distribution, I want to show that $\frac{d}{dx}E[a|a<x]<1\;\...
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1answer
40 views

How can I solve $\beta^2=\frac{m^2g}{h}\left(-\frac{\beta t}{m}+e^{\frac{\beta t}{m}}-1\right)$ for $\beta$?

This equation arose when I tried to find out how to derive $\beta$ in Stokes' Drag Force $F=\beta v$ as a function of the time $t$ it takes a mass $m$ to hit the ground after falling from a height $h$:...
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2answers
169 views

How to solve an equation with a tangent divided by a logarithm?

Here is an equation and I've never met this kind before. I would greatly appreciate your help. Maybe it's ridiculously simple and I overlook something? $$-12=\frac{\tan(x+4)}{\log(x+0.25)}$$
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1answer
26 views

Two Poisson r.vs with different rates may have same value at some argument

I am curious whether two Poisson distributions with two different rate parameters may have the same probability value at some positive integer argument i-e I am trying to solve $$\frac{e^{-\lambda _0}...
4
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2answers
225 views

Exponential and ln function

It seems quite simple but how would I find the exact solution for: $$\exp(x) = -\ln(x) $$ I'm not too sure where to start?
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1answer
38 views

When is $a(z) = b(c(z)) $?

Let $a(z)$ be a given transcendental entire function. When is $a(z)=b(c(z))$ where $b,c$ are also transcendental entire functions ? How to find such $b,c$ ? In particular when $a$ is given by a ...
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58 views

solve $a\cdot e^{b\cdot x}+c\cdot ln(x)=0$

Is it possible to find the analytical solution of $a\cdot e^{b\cdot x}+c\cdot ln(x)=0$? Is that a transcendental equation?
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68 views

Solving $-1=e^a-2e^{av}$ as part of a equation system

Problem Given $f_2(x)=e^{ax-b}+c$ with $x \in \left(0,1\right)$, I am trying to calculate the parameters $a,b,c$ in respect to the following constraints: $$ \begin{align} f_2(0) &= 0 \\ f_2(...
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66 views

What families of transcendental equations do we have solved?

I'm particularly interested in transcendental equations but searching in internet gives me only results about the classical linear-exponential equation (which is solved with Lambert's W) and its ...
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1answer
83 views

Transcendental equation $2 x n\cot (2x)= x^2 - n^2$

I have a transcendental equation and I have not a mathematical superiour formation (I'm an hydraulic engineer) necessary to solve it. The equation is : $2 x n\cot (2x)= x^2 - n^2$ or (same equation) :...
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1answer
39 views

How to find the Roots of the Derivative of two summed Gaussians.

Let $G$ be the Gaussian $$G(t,w,c,h)=w\cdot e^{-\dfrac{(t-c)^2}{2h^2}}$$ for some real parameter $t$ and the real constants $w$, $c$, and $h$. Now, let $F$ be a function defined in terms of $G$, $$F(t,...
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3answers
95 views

Simple Logarithms Equation

$$3^x = 3 - x$$ I have to prove that only one solution exists, and then find that one solution. My approach has been the following: $$\log 3^x = \log (3 - x)$$ $$x\log 3 = \log (3 - x)$$ $$\log 3 ...
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3answers
96 views

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 ...
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1answer
159 views

Solve $x^a = 1 - \exp(-x)$ for $x$

I would like to obtain a closed-form solution for the equation $x^a = 1 - \exp(-x)$, in which $x$ is the (real strictly positive) unknown and $a$ is a real positive parameter. So far, I have tried ...
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1answer
100 views

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): $$T=\frac{kV+g}{gk}(1-e^{-...
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51 views

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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39 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 ...
2
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0answers
71 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 ...