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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34 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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1answer
34 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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0answers
28 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 ...
2
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0answers
29 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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0answers
21 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
23 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 ...
2
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1answer
58 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 ...
2
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3answers
40 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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2answers
30 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
79 views

Solving a transcendental equation

How do I go about solving the following equation? $$x = A + B \log\left( \cosh\left(\frac{x}{C}\right)\right)$$
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1answer
41 views

How to solve the non-linear equation $-(a+c\,e)\left(\exp(-b/(a+c\,e))-1\right)-c\,d=f$ for $c$?

I have this non linear equation: $$-(a+c\,e)\left(e^{-\frac{b}{a+c\,e}}-1\right)-c\,d=f$$ The only unknown is $c$. All the coefficients ($a$, $b$, $c$, $d$, $f$) are real non-null costants. How can I ...
1
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0answers
31 views

Find the positive root of the equation $ce^{-c}-2(1-e^{-c})^2=0$

Can you help me find a root for $c$ in the equation below? $$ce^{-c}-{10\over5}(1-e^{-c})^2=0$$ By expanding this I got, $$ce^{-c}-2 + 4 e^{-c}-2e^{-2c}=0$$ now grouping, ...
0
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1answer
58 views

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, ...
1
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1answer
40 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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1answer
45 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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1answer
39 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
27 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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1answer
74 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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0answers
11 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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0answers
27 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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1answer
38 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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0answers
43 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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1answer
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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0answers
28 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 ...
3
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3answers
60 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
50 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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4answers
85 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
100 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
53 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
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1answer
118 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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0answers
166 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( ...
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1answer
38 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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1answer
46 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
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 ...
3
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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
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1answer
59 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
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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
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1answer
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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0answers
137 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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0answers
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, ...
6
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0answers
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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votes
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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1answer
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( ...
0
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0answers
27 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
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 ...
3
votes
1answer
131 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
182 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 ...