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Questions tagged [transcendental-equations]

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 solve this equation analytically

when considering the problem of solving how much mass of fuel does a rocket need in order to leave earth, we come across this equation: $$ v_{\infty}=-\frac{g}{Q}m_c+u\ln(1+\frac{m_c}{m_f}) $$ Where $...
realreal's user avatar
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Prove $\sqrt2$ satisfies a logarithmic equation

In a recent answer, I found that this definite integral takes on an exact value of $$\begin{align*} & \frac{4(a-1)}{\sqrt{(a-1)^2+1}} \left(a \sinh^{-1} \sqrt{a-1} - \sinh^{-1}(a-1)\right) \\ &...
user170231's user avatar
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8 votes
5 answers
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Approximate solution of $x^x=(x-n)^{(x+n)}$

Interested by this problem which ask for the solution of $$f_n(x)=x^x-(x-n)^{x+n}$$ that I rewrote as $$g(x)=x\log(x)-(x+n)\log(x-n)$$ After two series expansions, I obtained, for large $n$, as a very ...
Claude Leibovici's user avatar
14 votes
4 answers
524 views

Is $x^x = (x-1)^{x+1}$?

Background: I was trying to estimate the size of $21^{21}$ for some problem and decided to use $20^{22}$ as hopefully a rough approximate ($20^{22} = 2^{22} \cdot 10^{22} \approx 10^{28}$). But then I ...
mpear617's user avatar
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Using the r-Lambert function to solve a system of transcendental equations

I am trying to use the r-Lambert function applied to a vector in order to solve a system of transcendental equations, however, I am facing some difficulties when trying to obtain the right expression ...
Ignacio Canabal's user avatar
1 vote
1 answer
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A limit in a transcendental relationship [closed]

$$\sin((\frac{2\sqrt{x^2+1}}{r}+x)\pi)=(\frac{2x}{r\sqrt{x^2+1}}+1)\sin(x\pi),\ 0\le x\le1$$ I'm encountered with this equation when trying to maximise another expression. I want to know what is the ...
David Li's user avatar
1 vote
2 answers
126 views

Analytical Proof for the Inequality of Minimum Values of a Transcendental Function with Varying Number of Variables

Consider the following function: $$f_k(p_1,p_2,..,p_k) = \frac{2}{1 + d}\left((-1)^k + 2\sum_{i=1}^k (-1)^{i+1}p_i\right)\left((-1)^k + 2\sum_{i=1}^k (-1)^{i+1}p_i^{d+1}\right)$$ where $k$ is the ...
Amirhossein Rezaei's user avatar
3 votes
0 answers
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Location of zeros of quasi-polynomials

I'm considering quasi-polynomials of the form $$ P(z) = z^n + a_1z^{n-1}+...+a_n + K_1e^{-z\tau}(z^m + b_1z^{m-1}+...+b_m) + K_2e^{-2z\tau}(z^j + c_1z^{j-1}+...+c_j), $$ where all the constants are ...
Galois's user avatar
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Images of a vector under the Galois differential group span the solution set

I am reading the paper "A refined version of the Siegel-Shidlovskii theorem" by F. Beukers. In the proof of Theorem 1.5, he mentions the following results in Galois differential theory. Let ...
Khainq's user avatar
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3 votes
2 answers
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Calculating radius given an arc length and the distance between its endpoints

I wanted a formula that can tell the radius of the circle given the length of an arc of the circle and the distance between the endpoints of the arc. I have derived it to this form: $$d^2 = 2r^2\left(...
Prem Poddar's user avatar
1 vote
4 answers
146 views

Number of real solution of $2^{\sin(x)}-2e^{-\sin(x)}=2$ is

Number of real solution of $\displaystyle 2^{\sin(x)}-2e^{-\sin(x)}=2$ is What I try :: Put $\displaystyle 2^{\sin(x)}=t$, Then $\displaystyle t-\frac{2}{t}=2\Longrightarrow t^2-2t-2=0$ $\...
jacky's user avatar
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Solution for $(4+2r) x^{(1+r)}−x−1=0$

I am trying to find an analytic solution for the following equation: $$(4+2r) x^{(1+r)}−x−1=0$$ for $$r \in \mathbb{N} $$ and $$\frac{1}{2}<x<1, \space x \in \mathbb{R}$$ I am trying to solve ...
Amirhossein Rezaei's user avatar
2 votes
3 answers
110 views

Find $x$ such that $\cosh(a + bx) + 1 = cx$

I need to find an analytical solution for $x$ to: $$ \cosh(a + bx) + 1 = cx $$ where a,b and c are real parameters. I have tried to tackle this geometrically, by splitting the problem into finding ...
Gabriele Vecchio's user avatar
2 votes
1 answer
484 views

Solving Sine Equation [closed]

I'm trying to find all the real solutions of the following trigonometric equation : $$ \frac{\sin 3x}{\sin 2x} = A $$ I can see that $x$ is some integer function of $\pi$ , but I cannot find the exact ...
WizardLizard's user avatar
2 votes
1 answer
107 views

Are all solutions f(x) for f(x) = f(cos(x)) constant?

Under a post in this forum, I found a comment (https://math.stackexchange.com/a/46936/1173827) by Beni Bogosel that mentions the problem of finding all solutions for a function $f:\mathbb{R}\...
Space junk's user avatar
4 votes
1 answer
129 views

Can we solve $A \cos (\theta + \alpha) = \sin 2 \theta$ where $\theta \in [0, 2\pi)$ and $A, \alpha$ are constants?

What are solutions to $$A \cos (\theta + \alpha) = \sin 2 \theta$$ where $\theta \in [0, 2\pi)$ and $A, \alpha$ are constants? Graphically, this seems to have 2, 3, or 4 solutions, but I don't know ...
SRobertJames's user avatar
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Exact sum rule for solutions of the transcendental equation $x \tan x=\xi$

Consider the transcendental equation \begin{equation} x_j \tan x_j=\xi \end{equation} with $\xi$ real and positive and with $j=0,1,2,\dots$. The roots lie approximately at \begin{equation} x_j \...
adriaanJ's user avatar
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Are there any complex solutions to the equation ${x}^{2^x} = {2}^{{x}^{{x}^{2}}}$? [closed]

Thought of this question after learning about the Lambert W function and wanted to challenge myself. Are there any complex solutions to the equation $${x}^{2^x} = {2}^{{x}^{{x}^{2}}}$$ Tried to work ...
number eight's user avatar
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4 answers
128 views

Finding $a$ such that $\tanh(x)-a\sin^2(x)=0$ has a double root

Given $f(x) = \tanh(x)-a\sin^2(x)$, what is the value of $a$ for which $f(x) = 0$ has a double root, and what is the value of that double root? My Work : Using MAPLE , I plotted a few graphs and ...
Michael Jones's user avatar
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1 answer
147 views

Simplifying a complicated transcendental equation

I want to solve for $x$ the equation $$ x^{\alpha} (1-x)^{1-\alpha} = (\gamma x)^{\beta} (1-\gamma x)^{1-\beta} $$ where $\alpha,\beta,\gamma, x$ are all strictly between zero and one. If I'm not ...
raving-bandit's user avatar
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2 answers
123 views

How would you solve $3^x = 2x + 3$ using the Lambert $W$ function

Could someone provide a solution to the equation $$ 3^x = 2x+3. $$ Our teacher told us to solve it graphically, but I was curious what the exact answers might be and just plugged it into Wolfram ...
Norbert Domokos's user avatar
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0 answers
44 views

Solve $x = -(e^{Bx}z-y)Be^{Bx} z$ for $x$ or show existence or uniqueness of solution.

Let $ B,y,z \in \mathbb R$. Is it possible to solve the equation \begin{align} x = -(e^{Bx}z-y)Be^{Bx} z \end{align} for $x \in \mathbb R$? Or can one show that a unique solution $x \in \mathbb R$ ...
Jacob Körner's user avatar
1 vote
2 answers
145 views

Solving $\frac{8^x-2^x}{6^x-3^x}=2$

$$\dfrac{8^x-2^x}{6^x-3^x}=2$$ It is easy to see that in the domain of $\mathbb{R}\setminus\{0\}$, the solution is $x=1$. https://www.desmos.com/calculator/dsei8j2sdq. Desmos adds that the only one. ...
Fty56's user avatar
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6 answers
464 views

How can we find Lambert W solution to $\dfrac {x\ln x}{\ln x+1}=\dfrac{e}{2}$?

Find all real solutions: $$\frac {x\ln x}{\ln x+1}=\frac{e}{2}$$ Cross multiplication gives $$2x\ln x=\ln (x^e)+e$$ I didn't see any useful thing here. I tried solving this equation in WA. The ...
hardmath's user avatar
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2 answers
181 views

Exact solution to the the equation $(2 \pi - \theta)\cos \theta + \sin \theta = 0$

I have been trying to solve the following goat grazing problem: A goat is tied to the outside of a circular fence. If the length of the rope is the same as the circumference of the fence, what is the ...
russell.price's user avatar
3 votes
0 answers
301 views

Find $\int_0^\frac\pi2e^{i(ut+v\cos(t))}dt$ or $\int_0^\frac\pi2\sin(w+ut+v\cos(t))dt$ to invert $\frac{\sin(x)}x$

The solution to $\operatorname{sinc}(x)$$=a,0<a<\frac 2\pi$ involves inverting $ax-\sin(x)$ near $x=\frac\pi2 $ by transforming into $f_a(x)=a\left(x+\frac\pi2\right)-\sin \left(x+\frac\pi2\...
Тyma Gaidash's user avatar
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1 answer
109 views

Can anyone enlighten me on $e^{-x} = \alpha x$

I have worked for quite a while on a statistical problem and has been able to simplify the problem to an equation with a variable I have been seeking, $q$. The equation is $$ e^{-q} = \alpha q $$ ...
Anton's user avatar
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6 votes
2 answers
181 views

Convert $\frac1b\sum_{n=1}^\infty\frac{(b e^a)^n}{n!}B_{n-1}(an)$ to integral using $B_n(x)=\frac{n!}{2\pi i}\oint\frac{e^{x(e^t-1)}}{t^{n+1}}dt$

$\def\B{\operatorname B}$ In How to solve $x^{y^z}=z$ A solution uses Bell polynomials $\B_n(x)$ $$e^{ae^{bz}}=z=1+\frac1b\sum_{n=1}^\infty \frac{(ae^b)^n}{nn!}\B_n(b n)=\frac1b\sum_{n=1}^\infty\...
Тyma Gaidash's user avatar
6 votes
3 answers
641 views

Closed form for zeros of a function

I need a closed form for the zeros of $$f(x)=2\sin\left(\frac{\pi}{6}-\frac{\sqrt{3} x}{2} \right)-e^{-\frac{3x}{2}} $$ Putting $x=0$, we see that $f(0)=0$. For the closed form of the remaining zeros ...
Max's user avatar
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0 votes
1 answer
69 views

How to solve $x^x = n$ algebraically? [closed]

It can be solved graphically using the intersection point of $y= \log_ x (n)$ and $y=x$. For $n=2$ the intuitive solution is easy its $x=2$ only but how to solve it more generally for $n \in \mathbb{R}...
AdarW's user avatar
  • 23
2 votes
2 answers
84 views

How to solve $\,A\sin(\theta_2-\theta_1) - B\sin(\theta_1) = 0$

I want to find the solutions of the following equation (In order to find the singular points of a robot). $A,B$ are positive numbers, and actually : $A = 0.2531,$ $B = 0.2455.$ $$ A\sin(\theta_2 - \...
MIKE PAPADAKIS's user avatar
0 votes
1 answer
92 views

Find the condition such that $A \cos{x} = x$ has exactly two solutions.

This is something we all do in high school but I forgot how to solve such a problem. It recently came up in my theoretical Physics research. I want to find a constraint on the variable $A$ such that ...
SlothForeva's user avatar
3 votes
3 answers
453 views

Closed-form solution to the transcendental equation

Could you give me advice on how to find a closed-form solution $t>0$ to the following transcendental equation: $$(t+1)^a - t^a = g$$ where $a>1$ and $g>1$. An accurate closed-form ...
Piotr's user avatar
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1 vote
0 answers
103 views

How can you determine if a transcendental equation has elementary solutions or not?

I want(ed) to analytically solve the transcendental equation $e^{\sin(x)} = \sin(e^x)$ for a closed form solution. My working so far is: $\frac{e^{ie^x}-e^{-ie^x}}{2i}=e^{\frac{e^{ix}-e^{-ix}}{2i}}$ $...
user avatar
1 vote
1 answer
217 views

Verifying a Fourier series inverting $\frac{\tan(y)}y$ with the Whittaker W function

$\def\k{\operatorname k}\def\W{\operatorname W}$ Using a Dirac $\delta(x)$ Fourier series and this post $$\begin{align} y\cot(y)=\frac1x\implies\frac1{\sec^2(y)-x}=\int_0^\frac\pi2\delta(\tan(t)-xt)dt-...
Тyma Gaidash's user avatar
-2 votes
1 answer
94 views

Find $x$ in the exponential equation $3^x+4^x+5^x=6^x$ [duplicate]

$3^x+4^x+5^x=6^x(R:x=3)$ I try but I can't finish $3^x+2^x~2^x+5^x=3^x.2^x$ $3^x\cdot2^x-3^x-2^x\cdot2^x=5^x$
peta arantes's user avatar
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2 votes
0 answers
165 views

Closed form of $\int_0^{\pi\text{ or }\frac\pi2}\cos(w (\cos(t)+a t-b))dt$.

Although an integral for $x=\dots$ exists, it is slightly harder to integrate. Dirac $\delta(t)$ helps solve $\cos(x)+ax=b$: $$\frac1{\sin(x)-a}=\int_a^b \delta(\cos(t)+at-b)dt\tag1$$ From numerical ...
Тyma Gaidash's user avatar
0 votes
0 answers
55 views

On the Hurwitz stability of quasipolynomials

Suppose that $p,q\in\mathbb{R}^{+}$ and $a\in\mathbb{R}$. Consider the transcendental polynomial $$ p\lambda^2+p\lambda-a\left(e^{p\lambda}-1\right)e^{-q\lambda}=0,\;\lambda\in\mathbb{C}. $$ I would ...
user775349's user avatar
3 votes
4 answers
247 views

Is there a way to simplify $a\sqrt{1-a^2} + \arcsin(a) = \pi/4$?

A while ago, I was eating pizza and wondered that if you were to cut parallel to one of the radii, how far along would you need to cut in order to split a slice's area in half? In attempting to find a ...
Napthus's user avatar
  • 189
0 votes
2 answers
185 views

Find the all explicit real roots of $x^{1/x}\ln x^2=x.$

Using standard mathematical functions, find the all real roots of the equation: $$x^{1/x}\ln x^2=x.$$ I saw this question in the group of students studying mathematics. I tried to solve the equation ...
User's user avatar
  • 1,659
3 votes
2 answers
268 views

Is the number satisfying $\eta=\sin(\cos(\eta))$ transcendental?

I was graphing the function $\sin(\cos(\sin(\cos(\sin(\cos...$ when I realized it started to flatten out. This meant that this approaches a constant. Since the sine and cosine repeat, we can make a ...
Kamal Saleh's user avatar
  • 6,549
1 vote
1 answer
106 views

Generalised Lambert W and irreducible polynomials

I want to find the root of a function $f$ defined as $$ f(x)= e^{-cx} - \frac{P_n(x)}{Q_m(x)}$$ where $x,c$ are real numbers and $P_n,Q_m$ are irreducible polynomials of rank $n$ and $m$ respectively, ...
John Ritz's user avatar
4 votes
1 answer
255 views

The maximum number of intersections for a type of exponential function

Given two real valued functions $f(x)$ and $g(x)$ which both satisfy the conditions that they are sums of exactly $k$ positive terms each, each term of the form $b_i^x$ where $b_i \in \mathbb{N}$ $(i\...
user avatar
10 votes
3 answers
532 views

What is the reason that equations such as $\tan x = 2x$ can only be solved with the help of algorithms?

This is my first StackExchange question: What is the reason that equations such as $\tan x = 2x$, $\cos x = x$, $\sin(x) = x^2$ and other questions that involve the same variable within a ...
user avatar
8 votes
4 answers
295 views

Approximate inverse of $k=\frac{\log (1-t)}{\log (t)}$

Trying to answer this question where we look for the solution of $$\large\color{red}{t^k+t=1} \qquad \qquad \text{with} \qquad \color{red}{0<k<1}$$ which is more or less the function Lambert ...
Claude Leibovici's user avatar
2 votes
1 answer
123 views

Solution to transcendental equation

I'm looking for solutions to the equation $$x^2+2^x+x^x = 12$$ Which is satisfied obviously by $x=2$ and somewhat less obviously by $x\approx-3.4512$. By plotting $|z^2 + 2^z + z^z - 12|$ on the ...
MukundKS's user avatar
  • 162
0 votes
1 answer
121 views

Dominant Balance of a Transcendental Equation

Consider real roots to the equation $$ \frac{2}{1- \varepsilon x^{2}} = e^{x} $$ as $\varepsilon \to 0$. From the monotonicity of $e^{x}$ together with qualitative properties of $\frac{2}{1-\...
Ron Shvartsman's user avatar
3 votes
2 answers
97 views

How can this equation be simplified to give $y$?: $x = \frac{(-1)^y ( 5 (-1)^y y - y + (-1)^y - 1))}4$

I'm trying to convert this equation to the form $y = ...$, but I am stuck. It seems the $y$-root of $(-1)^y$ is not $-1$, but is instead a beast. Here is the overall equation: $$x = \frac{(-1)^y ( 5 ...
CommaToast's user avatar
1 vote
1 answer
133 views

Finding the value of $\sin^3x+\cos x$, if $x$ is an acute angle satisfying $2\sin x\sin\left(\frac{x}{2}\right)=1-\sin x$

Given that $x$ is an acute angle satisfying $$2\sin x\sin\left(\frac{x}{2}\right)=1-\sin x$$ Then find the exact value of $$\sin^3 x+\cos x$$ My try: Letting $x=2t$ we get $$2\sin(2t)\sin(t)=1-\sin(2t)...
Ekaveera Gouribhatla's user avatar
1 vote
1 answer
73 views

Cartesian equation for a transcendental / trigonometric curve

Hello! Please see figure above. I am searching for the cartesian equation for the curve in green, similar to how the equation for a semicircle is $f(x) = √(1 - x^2)$. I'm not sure if this is even ...
VinMilligan's user avatar

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