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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Number of solutions of a transcendental function

I am studying time-delay differential equations now. When I read "textbook" material, my teacher wrote it, this lemma occurs to me. For a transcendental function, ...
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34 views

How can we solve the “transcendent” equation relating to Stoner criterion

I met a algebraic equation(not a transcendent equation) during my study of Stoner criterion in Quantum Statistical Physics. In this occasion, one need to solve the equation $$ ...
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44 views

Relationship between constants in an equation

I have the following equation: $e^{ax} + e ^{bx} = e ^{cx}$ Is it possible to find a relationship between constants $c$ and $a$, $b$ that holds for all $x$'s? Thanks in advance.
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31 views

Generalization of Lambert W function?

Can the function $f(x)$ defined by $$ x = f(x)^2 e^{f(x)}$$ for real $x>0$ be expressed in relation to the Lambert W Function?
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1answer
24 views

How to solve transcendental hyperbolic equation

How can I solve the functional relation $$ e^{-af'(x)}\cosh( f(x) ) = bx $$ for $f(x)$? It would suffice to solve for $x>0$, $a>0$ and $b>0$.
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71 views

How does one analyze $\phi = \beta \sin\phi$?

Consider the following transcendental equation: $$\phi = \beta \sin \phi . \qquad (*)$$ How does one generate a description of how $\phi$ depends on $\beta$? My attempt From inspection (i.e. ...
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9 views

Choice of bounds for functions “defined” as integrals using the FTC

Lately I have been watching (for personal enlightment) the MIT Open Courseware course of single-variable calculus, which is diving into things that go way, way beyond what my high school calculus ...
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25 views

Solving a transcendental function using the Lambert function

The solution to the equation $$Xe^X=K$$ is given by $$X=W(K)$$ where $W$ is the Lambert function. This idea was extended here to show that the solution to $$\frac{1-e^X}{X}=K?$$ is given by ...
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37 views

Why should transcendental functions and their arguments be dimensionless?

While looking for the answer on the internet I came across an answer giving this explanation "Another way of seeing clearly why an exponential's argument should be dimensionless is to Taylor expand: ...
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42 views

Stability of transfer functions with internal delay

I would like to know what the best method is for finding stability of transfer functions that have internal delays. Basically I have a transfer function of the form: $\frac{f(s) e^{-st}}{g(s) + h(s) ...
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20 views

Plotting a Transcendental Function

How would I plot $m_{2}$ as a function of $m_{1}$ for $0.5< m_{1} < 10$, for the following equation: $$\frac{\sin(m_{2})}{(m_{1}+m_{2})^{2}} = \alpha,$$ where $\alpha$ is a non-zero constant? ...
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27 views

Asymptotic parameter for a transcendental equation

I need to find the roots of the following equation $(x^2-a^2)(x^2+a^2)\sin(b^2-x^2)-b^2 \cos(\sqrt{b^2-x^2})=0$. Say $\mathcal{A}=(x^2-a^2)(x^2+a^2)$. I assume that as long as $x$ is away from its ...
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1answer
36 views

Finding roots to transients

I understand how to solve 2 element transient but am having some problems with 3+ element transients. Specifically I'm trying to solve this equation: $$737280 e^{-2400t}-576000 e^{-1500t} + 46080 ...
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0answers
45 views

Solve $z+\sin{z}=i$

How can I find how many solutions following equation have? $$z+\sin{z}=i$$ I can make substitution $z=it$ and get $$t+\sinh{t}=1$$ which has one real solution $t\approx0.4900730685$ thus ...
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3answers
59 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 ...
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0answers
49 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 ...
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84 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 ...
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2answers
99 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 ...
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52 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} ...
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1answer
117 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 $a$ ($a$ ...
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145 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 ...
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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( ...
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1answer
36 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 + ...
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45 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 ...
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38 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 ...
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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 ...
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1answer
57 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 ...
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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 ...
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75 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 $V=\tanh(x)$ ...
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136 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. $\pi^a = e^b$ ?
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28 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, ...
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40 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?
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1answer
74 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.
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107 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( ...
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26 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
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1answer
43 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 ...
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1answer
128 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 ...
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2answers
171 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 ...
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5answers
137 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 $y(2) = 4$ ...
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1answer
109 views

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

I have the following equation of $t$: $\text{C0}+(\text{C1}+\text{C2} t) \cos (\text{C4} t)+\sin (\text{C4} t) (\text{C7}+\text{C8} t)+\text{C5} \cos (\text{C6} t)+\text{C9} \sin (\text{C6} t)=0$ ...
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42 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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65 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.
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1answer
81 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?
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2answers
50 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 ...
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119 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 ?
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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 ...
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34 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 ...
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236 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
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191 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 ...
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1answer
64 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 $$\sin{x}\approx x-\dfrac{1}{6}x^3$$ ...