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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3
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55 views

May $y=e^x$ be satisfied with both $x$ and $y$ $\in$ $\mathbb{Z}^+$?

May $y=e^x$ be satisfied with both $x$ and $y$ be positive integers? I think it is not possible as $e$ ,a transcendental number, when multiplied by itself would never result in rational number. Am I ...
0
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0answers
26 views

Need to find two unknown values in given equation.

$$I = I_L - I_0 \left( \exp \left( \frac{q(V-IRs)}{nkT} \right) -1 \right) - \frac{V-IRs}{R_p}$$ I want to find the values of two unknown variables in given equation i.e $I$ and $V$. I know that I ...
0
votes
4answers
78 views

How to solve $\log n = \frac{\log 2}{10} \sqrt{n}$

I need to solve $\log n = \frac{\log 2}{10} \sqrt{n}$. I know it is a transcendental function and also hear about generalizes Lambert function (Lambert W-function) could help me to solve it. But I ...
1
vote
2answers
87 views

Roots of transcendent equations $\tan{x}=bx$ and $x\tan{x}=b$

We know that transcendent equations $$\tan{x}=bx$$ and $$x\tan{x}=b$$ can not be solved exactly. But what I concerned most is the relationship between their non-trival roots $x_{n}^{(1)}$ and ...
2
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1answer
43 views

Limit solution to a transcendental equation

Let $n\ge 1$ be a positive integer. The question is to solve the following transcendental equation: \begin{equation} \left(1+q\right)^{2 n} = \frac{\sqrt{\pi}}{2} \frac{1-q}{\sqrt{q}} \sqrt{n} ...
2
votes
1answer
105 views

Solving equation $a^{-x} + \log x/\log a = 0$

Please can you instruct me how should I start writing an algorithm (pseudo-code, to be implemented) for finding all solutions for the following equation: $a^{-x} + \log x/\log a = 0$ where $a$ ($a$ ...
0
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0answers
75 views

Can't find solution to Calculus 8th (Adams, Essex) problem

I've been sitting here for hours trying to find a solution to his problem. If you have the function $g(y)$, which is the inverse of $f(x) = x^x,\\ e^{-1} \leq x < \infty,$ show that ...
2
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0answers
37 views

Functional Equation involving derivatives and time-steps [duplicate]

I am attempting to solve the equation $$f(x + 1) = f'(x)$$ for distributions $C \rightarrow C: f(x)$ My first guess to exploit the fact that this seems similar to identity $$\sin\left( ...
1
vote
1answer
29 views

Solving a second level functional equation over all functions $g$

I am trying to find a closed form expression $f$ such that $$f(g(x+1) - g(x)) + f(g(x) - g(x-1)) = f(g(x))$$ For all functions $g$ I have concluded that for polynomials $$2^{n+1}f(0) = f(a_0 + ...
-1
votes
1answer
42 views

Explicit solution of the following equation possible?

Is it possible to obtain an explicit solution for $K$ for the following equation? $$(e^K - 1)(e^{\beta K} - 1) = q$$ for $0\leq q \leq 4$ and $0\leq \beta \leq 1$ For $\beta=1$ one gets ...
2
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0answers
37 views

Interpolation of iterated logarithms

$$\text{Let }\log^2(x)=\log(\log(x)),\\ \text{ then }f(y,x)=\log^{\lfloor1+y\rfloor}\left(\log(x)/\log((1-x^{1/x}(y-\lfloor y\rfloor))+(y-\lfloor y\rfloor))\right)$$ gives an interpolation between ...
3
votes
0answers
30 views

$\log_j(\log_j(\log_j(x)))=\log(x);\ \ j=?$

$\log_j(\log_j(x))=\log(x)$ has solution $j=x^{\exp-W(\log^2(x))}$ for real $x\neq0$, where $W=$ Lambert W function. But what is the solution to $\log_j(\log_j(\log_j(x)))=\log(x)$? Mathematica can't ...
0
votes
1answer
55 views

Bounds and uniqueness of a transcendental equation

Let $p\in[0,1]$ and $\rho(x): [0,1] \rightarrow [0,\infty)$ such that $$\int_0^1 dx \rho(x) = 1.$$ I'd like to investigate the following transcendental equation: $$\frac{1}{2p} = \int_0^{1} dx ...
0
votes
1answer
56 views

Thanks to what I'm able to reduce analytic functions in algebraic form?

Usually I take this for granted, but lately I had an encounter with some infinitesimal calculus concepts from a computational point of view, Fourier transformations for the most part, and I can't wrap ...
4
votes
1answer
74 views

Find general solution of first order non-linear in a transcendental function

I have the function $$\frac{dV}{dT}=1-V^2$$ Just looking to see if my working is okay. $$dV=1-V^2dT$$ $$\frac{1}{1-V^2}dV=dT$$ Integrate $$\int{}\frac{1}{1-V^2}dV=\int{}dT$$ Let $V=\tanh(x)$ ...
5
votes
0answers
132 views

Are there integers $a, b$ s.th. $\pi^a = e^b$?

Is $\log \pi $ a rational number? That is, are there non-zero integers $a, b$ s.th. $\pi^a = e^b$ ?
0
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0answers
27 views

nonlinear algebraic equations

Could we obtain "some information" about the profiles of u=u(x) and v=v(x) satisfying the following equations: d1*Log u+a1*x+g11*u+g12*v=c1, d2*Log v+a2*x+g21*u+g22*v=c2, where d1, d2, a1, -a2, ...
6
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0answers
37 views

Inverse image of rationals under tangent function is free abelian?

It is easy to see that the set $\{x:\tan x\in \Bbb Q \,\, or\,\, \pm\infty\}$ forms a group under addition. It is a free abelian group?
0
votes
1answer
70 views

How to prove $f(x)=5\sqrt{x^4+1}$ is a transcedental function

$$ f(x)=5\sqrt{x^4+1} $$ I know this function is transcedental function which is also not algebraic function, but i'm not sure how to prove this. Thank you for anyone to help.
0
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1answer
103 views

How to solve that equation?

How to solve the equation $$ \left| \tan \left( x \right) \tan \left( 2\,x \right) \tan \left( 3\, x \right) \right| + \left| \tan \left( x \right) +\tan \left( 2\,x \right) \right| =\tan \left( ...
0
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0answers
25 views

Approximately minimising a transcendental function.

I currently have a closed form solution for the error probability of a certain type of wireless channel. By letting all $S_i$ terms denote constants, using $U(\cdot,\cdot,\cdot)$ to denote the ...
2
votes
1answer
41 views

What kind of algebraic equations do trandescendal numbers not solve?

I know transcendental numbers cannot solve polynomials or rational functions (since they can always be written as a polynomial), but are they the solutions to equations containing a variable raised to ...
3
votes
1answer
113 views

Is tetration a transcendental function?

Is tetration a transcendental function? If so are there any papers with a proof? I suspect that it is because I have not seen any algebraic situations where tetration is the answer and the fact that ...
0
votes
2answers
110 views

solve equation of erf

I'd like to solve this equation for $\mu$. Is it possible? If not, why? $$ 2 P = \operatorname{erf}\left( \frac{\mu - A}{ \sqrt{2 \sigma^2} } \right) - \operatorname{erf}\left( \frac{\mu - B}{\sqrt{2 ...
5
votes
5answers
130 views

Is there analytic solution to $x^y=y^x\land x\neq y$ as $y(x)$?

Equation $x^y=y^x\land x\neq y$ has trivial solution $ y(x) = x$. Is there non trivial solution given say in terms of elementary or special functions as $y(x)$? A solution that would yield $y(2) = 4$ ...
0
votes
1answer
95 views

Algebraically find roots of a function composed of linear equations and trigonometric functions

I have the following equation of $t$: $\text{C0}+(\text{C1}+\text{C2} t) \cos (\text{C4} t)+\sin (\text{C4} t) (\text{C7}+\text{C8} t)+\text{C5} \cos (\text{C6} t)+\text{C9} \sin (\text{C6} t)=0$ ...
0
votes
1answer
41 views

Solve differential equation solution?

I want to solve for $Y(x)$: $$ Y(x) = A - Bx + C\ln(A/Y(x)) $$ where $A$, $B$, and $C$ are defined. Not sure how to go about this. I'm tempted to treat $x$ and $Y(x)$ independently and solve them ...
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3answers
64 views

Finding the unknown variable

What is the value of $x$ in $x^{x}=25$? How can this be solved in the easiest way of all? I just couldn't deduce any idea regarding where to start.
1
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1answer
76 views

Solution for Transcendental Equation $B+x\sin{\left(\frac{A}{x}\right)}=0$

I am trying to solve a transcendental equation of the form: $$B+x\sin{\left(\frac{A}{x}\right)}=0,$$ where both $A$ and $B$ are constants. What would be the best approach to solve it?
2
votes
2answers
49 views

How to solve the transcendental equation $a^h=bh+c$ with a parameter

I've got a random Rayleigh variable $\xi$ with $p_\xi(x)=\frac{x}{\sigma^2}\exp\{-\frac{x}{2\sigma^2}\},x\geq0$ There are two hypotheses: $H_0:\sigma=\sigma_0$ and $H_1:\sigma=\sigma_1$ I have built ...
4
votes
3answers
115 views

How to find number of real roots of a transcendental equation?

The number of real roots of the equation $$2\cos\left(\frac{x^2+x}6\right)=2^x+2^{-x}$$ Another question is... can we use descartes rule of sign in here or in any transcendental equation ?
0
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0answers
121 views

How Unsolvable are Transcendental Equations

When working with Wein's displacement law, I came across a transcendental equation similar to this one: $e^{-x} + x = 1$. Reading about these on the internet, I'm a little confused: wikipedia defines ...
1
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0answers
30 views

Is there a closed form for the values $x$ where $f(x) = 0$ and when $f(x) = 1$

I posted an answer to the question An Integral Involving The Inverse Of $f(x)$ and my answer depends on knowing where the function $f(x)$ is $0$ or $1$. The function itself is $$f(x) = \log x - \log ...
6
votes
3answers
192 views

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 ...
5
votes
3answers
176 views

How do I prove this transcendental equation has a solution?

I am trying to prove that for the following equation, there is a B that solves it (c is a constant): $1-B = e^{-cB}$ I understand this is a transcendental equation, but how do I prove there is a B ...
1
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1answer
62 views

How find the equaition $x^2\sin{\dfrac{1}{x}}=2x-501$ root

find the equation approximate solution , such the root of $$x^2\sin{\dfrac{1}{x}}=2x-501$$ to an accuracy of $ 0.001$ I think this problem use this $$\sin{x}\approx x-\dfrac{1}{6}x^3$$ ...
0
votes
1answer
190 views

Find the smallest number b such that the function

Find the smallest number b such that the function $f(x)=x^3+7x^2+bx+4$ is invertible. Evaluate $\frac{\mathrm{d}}{\mathrm{d}x}(f^{-1})(4)$ using that $b$.
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1answer
138 views

How to solve one unknown equations like $15^x = x^{16}$ or $x^x = 1991$ etc…?

I am curious how to solve equations like $15^x = x^{16}$ or $x^x = 1991$. Yes, of course Wolfram Alpha or Matlab can calculate it, or it can be approximated (as we learned in school). But are there ...
0
votes
1answer
70 views

Given $\log(p(x)) = q(x)$ are $p$ and $q$ algebraically independent?

Since $e^x$ and $\log y$ are transcendental functions, does $$\log p(x) = q(x)$$ mean that polynomials $p$ and $q$ (of finite degree $n$ and $m$ respectively) are algebraically independent? What ...
0
votes
1answer
43 views

Analytical expression for the height of a circular segment

The area of a circular segment is $A=\frac{R^2}{2}\left(\theta - \sin\theta\right)$ Considering $A$ and $R$ known, can you find an analytical expression for $\theta$? Or am I forced to solve it ...
3
votes
0answers
114 views

Special functions useful for solving a transcendental equation

I have the following transcendental equation which may very well lack an analytic solution. I would, at the least, like an expression for the relationship between $\theta$ and $\phi$ in some closed ...
0
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2answers
145 views

Solving transcendental equation involving exponential functions

I'm trying to numerically solve the following transcendental equation: $$(\alpha x + \beta)=\delta e^{\gamma x}$$ with $\alpha$, $\beta$, $\gamma$ and $\delta$ real, positive constants. It is ...
3
votes
2answers
660 views

How to solve equations with logarithms, like this: $ ax + b\log(x) + c=0$

I encountered an equation of type $$ ax + b\log(x) + c=0$$ Here a, b, and c are constants. Does anyone know how to solve these type of equations? I guess this way: $$\log(x)= \frac{c-ax}{b}$$ $$x= ...
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0answers
68 views

Are the special functions independent?

maybe the bessel functions are some complicated function of the exponential function, logarithm function... or maybe there's a relation between two or more transcendental functions. Is there a way to ...
0
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0answers
91 views

Numerical methods for solving nonlinear equation

I need to solve some nonlinear equation which looks like this: $$x = \frac{L\cos(Wx)}{1+LW\sin(Wx)}$$ I have listened about some method of the bruteforce of the roots. Can you help me to find some ...
20
votes
1answer
326 views

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 $$x \approx 14.498719188878466465738532142574796767250306535...$$ ...
5
votes
1answer
172 views

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!
4
votes
5answers
850 views

How do i solve $e^{ax}-e^{bx}=c$ for $x$?

How do i solve $e^{ax}-e^{bx}=c$ for $x$? The constants $a$, $b$ and $c$ are real numbers. It is the final form of a longer equation that I simplyfied. Edit: The actual equation I'm trying to solve ...
1
vote
1answer
1k views

How to solve transcendental equations with MATLAB?

Here's the equation: $$ - \frac{MN}{\sqrt{2\pi \left( \sigma_1^2 + \sigma_2^2 \right)}} \frac d {dy} \exp \left( -\frac {y^2} {\left( \sigma_1^2 + \sigma_2^2 \right)} \right) = k \left( x-y \right) ...
0
votes
1answer
107 views

Location of roots of certain transcendental equations.

If $a$ and $b$ are two solutions of $\,e^x \cos x -1=0$, then how many solutions of the equation $ e^x \sin x-1=0$ lie between $a$ and $b$ ?