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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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 ...
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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 ...
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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 ...
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What is the exact value for the solution to $(\sin x)^{\cos x}=2$? [duplicate]
What is the exact value for the solution to this equation?
$$(\sin x)^{\cos x}=2 \tag1$$
A similar equation $$(\sin x)^{\sin x}=2 \tag2$$ can be solved using the Lambert's $W$ function. But with $(1)$,...
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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 ...
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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 $x \in \mathbb R$ ...
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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. ...
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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 ...
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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 ...
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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)$$=a,0<a<\frac 2\pi$ involves inverting $ax-\sin(x)$ near $x=\frac\pi2 $ by transforming into $f_a(x)=a\left(x+\frac\pi2\right)-\sin \left(x+\frac\pi2\...
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Can anyone enlighten me on $e^{-x} = \alpha x$
I have worked for quite a while on a statistical problem and has been able to simplify the problem to an equation with a variable I have been seeking, $q$.
The equation is
$$
e^{-q} = \alpha q
$$
...
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Study number of complex zeros of a function contained in a strip
Consider the equation $$8\pi e^{-2}z-e^{-2z}+4E_{1}(2z)=0$$ ($E_{1}$ is the exponential integral of order 1).
I would like to study the number of solutions in the strip $0<\Re(z)\le 1$ (and ...
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Evaluate $\frac{d^{n-1}}{dw^{n-1}}\csc^{-1}(w)^n$ or $\frac{d^{n-1}}{dw^{n-1}}\left(\frac{\sqrt{w^2-1}+i}w-1\right)^m$
With Lagrange reversion:
$$x\sin(x)=1\iff x=\{2\pi k+\csc^{-1}(x),(2k+1)\pi-\csc^{-1}(x)\}$$
Therefore:
$$x_{2k}=2k\pi+\sum_{n=1}^\infty\frac1{n!} \left.\frac{d^{n-1}}{dw^{n-1}}\csc^{-1}(w)^n\right|_{...
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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 $\B_n(x)$
$$e^{ae^{bz}}=z=1+\frac1b\sum_{n=1}^\infty \frac{(ae^b)^n}{nn!}\B_n(b n)=\frac1b\sum_{n=1}^\infty\...
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Is there a double series expansion of $\ln^n(\ln(a)+2\pi i n_2-\ln(w)),w=2\pi i n_1;n_1,n_2\in\Bbb Z$?
Problem:
Via Lagrange reversion with $2$ sets of branches:
$$f(z)=ze^{e^z}\implies f^{-1}_{n_1,n_2}(z)=2\pi i n_1+\sum_{n=1}^\infty\frac1{n!}\frac{d^{n-1}}{dz^{n-1}}\ln^n(\ln(a)+2\pi i n_2-\ln(w))\...
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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 ...
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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}...
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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 :
$A = 0.2531,$
$B = 0.2455.$
$$ A\sin(\theta_2 - \...
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Find the condition such that $A \cos{x} = x$ has exactly two solutions.
This is something we all do in high school but I forgot how to solve such a problem. It recently came up in my theoretical Physics research. I want to find a constraint on the variable $A$ such that ...
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Closed-form solution to the transcendental equation
Could you give me advice on how to find a closed-form solution $t>0$ to the following transcendental equation:
$$(t+1)^a - t^a = g$$
where $a>1$ and $g>1$. An accurate closed-form ...
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How can you determine if a transcendental equation has elementary solutions or not?
I want(ed) to analytically solve the transcendental equation $e^{\sin(x)} = \sin(e^x)$ for a closed form solution.
My working so far is:
$\frac{e^{ie^x}-e^{-ie^x}}{2i}=e^{\frac{e^{ix}-e^{-ix}}{2i}}$
$...
user1112591
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answer
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Verifying a Fourier series inverting $\frac{\tan(y)}y$ with the Whittaker W function
$\def\k{\operatorname k}\def\W{\operatorname W}$
Using a Dirac $\delta(x)$ Fourier series and this post
$$\begin{align}
y\cot(y)=\frac1x\implies\frac1{\sec^2(y)-x}=\int_0^\frac\pi2\delta(\tan(t)-xt)dt-...
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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
$3^x+2^x~2^x+5^x=3^x.2^x$
$3^x\cdot2^x-3^x-2^x\cdot2^x=5^x$
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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\...
user1114342
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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 ...
user1112591
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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 ...
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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 ...
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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-\...
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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:
$$x = \frac{(-1)^y ( 5 ...
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Finding the value of $\sin^3x+\cos x$, if $x$ is an acute angle satisfying $2\sin x\sin\left(\frac{x}{2}\right)=1-\sin x$
Given that $x$ is an acute angle satisfying $$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)...
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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 ...
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Help Me Minimize Freezer Burn on Ice Cream by Understanding Transcendental Equations
I'm a high school math teacher trying to stay engaged with the subject, and I've started wondering about this question:
"Given a cylindrical ice cream container with radius r and height h, ...
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How to extract $x$ from the exponential equation $(x-1)\cdot\left(2^{1/x}-1\right)=k$ when $k\in(0,1)$ and $x\gt 1$?
I spent quite a time searching the internet to find a way to extract $x$ from the exponential equation $(x-1)\cdot\left(2^{1/x}-1\right)=k$ when $k\in(0,1)$ and $x\gt 1$
I would appreciate any hints.
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Solving $a x = [\ln(x) - b]^c$
I'm trying to solve the nonlinear equation $a x = [\ln(x) - b]^c$ for x, where a, b, and c are constants, for a project. I've tried numerical techniques like in Excel Solver, but the solutions seem ...
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Rewriting continued fraction for spheroidal eigenvalue function $\lambda_{n,m}(z)$
I know little about spheroidal functions and browsing Wolfram functions, the Spheroidal Eigenvalue function $\lambda_{n,m}(z)$ was intriguing. According to the DLMF section 30.3(iii):
$$b_p-\lambda-\...
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When to stop looking for closed-form solutions?
I have the following equation which is simplified from another relatively long equation.
$$
xe^{-x}= c (\frac{x}{c})^{-x}
$$
Here Lambert W function doesn't help since you still get $x$ inside the ...
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Solving $\frac{2x^2}{x-1}\cdot 2^x+8=0$
So recently, a friend of mine in grade $12$ got this question on her homework.
$$\left(\frac{2x^2}{x-1}\right)\left(2^x\right)+8=0$$
I tried rearranging this expression into $$\frac{-4x+4}{x^2}=2^x$$
...
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How to get the closed form of the eigenvalue roots for the eigenvalue equation by using the Lambert W Function Approach.
The eigenvalue equation is
$\lambda^2 + \lambda a e^{-\lambda \tau_c} + b e^{-\lambda \tau_c} = 0$ where
$\lambda$ is a variable, and other parameters including $a \in R$, $b \in R$ and $\tau_c \in R$...
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5
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Can the equation $2^x + x = 11$ be solved without graphing?
$2^{x}+x = 11$
Well this problem is easy to solve just by looking at its graph, and we find the answer is $x = 3$, but I want a way of solving it rather than just looking at it to find the solution. I ...
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Is it possible to solve the equation $x - 1 = x^{-y}$ explicitly?
I'm trying to solve the equation
$$
x - 1 = x^{-y}
$$
or to find the inverse of the function that is represented by this equation - both explicitly (symbolically).
However, I cannot find a way to do ...
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2
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how do I solve this equation in the form $c$=$e^b+c^2e^c$
I only recently learned what the Lambert W function is and how to apply it to different problems. But this expression
$c=e^{-4cx}+c^2e^{-cx}$
is something that I was not able to solve using Lambert ...
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How do you get an expression for $x$ in terms of $c$ in the equation $\ln(x)=cx$?
I am in the middle of solving this problem and I came across a sub problem that sorta has the format
$$\ln(x)=cx$$
I tried looking online on how to solve for $x$ if given $c$ but I couldn't find ...
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Finding number of solutions to $\sin(x)=x/10$ using an algebraic method.
I am trying to find the number of solutions of the equation $\sin(x)={x/10}$. While I know about the graphical method of doing this, I want to know if there are any quicker and/or algebraic method to ...
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3
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How to solve $x^y = ax-b$
I have encountered this equation:
$$x^y = ax-b$$
I know to find $y$ as a function of $x$ then:
$$y\ln(x) = \ln(ax-b)$$
$$y = \frac{\ln(ax-b)}{\ln(x)}$$
or
$$y = \log_x(ax-b)$$
But the problem I need ...
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