Questions tagged [transcendental-functions]

Transcendental functions are those functions that do not satisfy an algebraic equation.

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Why isn't there an exclusive hypernym for exponential and logarithmic?

Mathematicians, transcendental is a very broad word. Why isn't there a narrow word just for exponential and logarithmic? Transcendental is compared to algebraic, but there is no hyponym of ...
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Indefinite integral involving product of sinc with cosine: $\displaystyle\int\operatorname{sinc}^2(ax) \cos(bx)dx$

I try to solve this indefinite integral: \begin{equation} \int\,\mathrm{sinc}^{2}\left(ax\right)\,\cos\left(bx\right)\,\,\mathrm{d}x \end{equation} whith $\mathrm{sinc}\left(x\right) = \frac{\sin\...
Dennis Marx's user avatar
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Why is $\ln2\approx 0.4^{0.4}$?

Just stumbled upon $$\ln2\approx 0.4^{0.4}$$ and wondered if that's just a coincidence, or whether there's some deeper reason? $$\ln2 - 0.4^{0.4}\approx 0.00000234$$ which is a relative error of just $...
emacs drives me nuts's user avatar
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Solving integrals involving ratio of transcendental functions

I am facing integrals of the form \begin{equation} I = \int \frac{\sin^{2}\left(x\right)}{\sin^{2}\left(x\right) + C\sin\left(2x\right) + C^{2}x^{2}}\, \mathrm{d}x \end{equation} for which I could not ...
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L'Hopital Rule: For what value(s) of $p$, if any, is this limit equal to $1$?

For what value(s) of $p$, if any, do we have $$ \lim_{x \to p} \frac{ p^x - x^p }{ x^x - p^p } = 1? $$ My Attempt: Since $0^0$ is indeterminate, we must have $p \neq 0$. If we put $x = p$ into $\...
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Transcendence of function and change of fields

Suppose the one has a sequence of rational functions $Q_n(z)\in\mathbb Q(z)$. Let $p$ be a prime number. Suppose that that there exists an infinite subset $X$ of $\mathbb Q_p$ such that: the ...
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Show that $\arcsin(\frac{x-1}{x+1})=2\cdot\arctan(\sqrt{x})-\frac{\pi}{2}$

So I started by saying that $$y=\arcsin\left(\frac{x-1}{x+1}\right)$$ Then you could say that $$\sin(y)=\frac{x-1}{x+1}$$ Then calculating $\cos(y)$ with the trigonometric identity, I found the ...
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Formal proof using complex analysis that $g(x)=e^{\frac{1}{\log x}}$ is not an algebraic curve?

I would like a formal proof using complex analysis that $g(x)=e^{\frac{1}{\log x}}$ is not an algebraic curve (in $\Bbb R^2$) I give my proposed proof by contradiction here: Consider the isometry: $$\...
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A transcendental function?

Let $(u_n)_n=(p_n/q_n)_n$ $\left(\text{with $\mathrm{GCD}(p_n,q_n)=1$ and }\ln|u_n|=o(2^n)\right)$ a sequence of rational numbers that is not ultimately zero. I want to prove that the function $f(z)=\...
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Is this proof that $e^z$ is transcendental correct?

Lemma. If $p(z)$ is a non-zero polynomial and $k\geq 1$ a natural number, then there exists a polynomial $q(z)$ of the same degree such that $\frac{d}{dz}\Big(p(z)\cdot e^{kz}\Big) = q(z)\cdot e^{kz}$...
Dave Moutardier's user avatar
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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 ...
VinMilligan's user avatar
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Do smooth "almost-algebraic" transcendental curves exist?

Is it possible for a smooth, transcendental plane curve to be such that for every point $(a,b)$ on the curve, there is a polynomial with rational coefficients $p(x,y)\in\mathbb{Q}[x,y]$ where $p(a,b)=...
semisimpleton's user avatar
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weird series notation ANS: Generalized Hypergeometric Functions

Hi I came across this curious notation on a twitter account (@infseriesbot) that supposedly posts series functions. The notation is very strange to me, hopefully some one who is familiar with this ...
TheSprintingEngineer's user avatar
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Evaluate $\zeta'{(3)}$ and $\zeta'{\left(3,\frac{1}{4}\right)}$

I am trying to evaluate this sum: $$\sum_{k=0}^{\infty}\frac{(-1)^k\log{(2k+1)}}{(2k+1)^3}$$ After doing the rest of works, there are still these two terms that I can not find the closed form: $\zeta'{...
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$F$ be a field and $F(x)$ be the field of rational functions in $x$ over $F$. Then the element $x$ of $F(x)$ is transcendental over $F$

Let $F$ be a field and $F(x)$ be the field of rational functions in $x$ over $F$. Then the element $x$ of $F(x)$ is transcendental over $F$. Proof: $F(x)$ is clearly a field extension of the field $F$....
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The composition of an algebraic function and a transcendental function

I say that a function $f : \mathbb{R} \to \mathbb{R}$ is algebraic if it is a solution of a polynomial equation, that is there exists a polynomial $F(x,y)$ such that $F(x,f(x)) =0$ for every $x$. I ...
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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 ...
thelordabdo's user avatar
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Solving system of transcendental equations algebraically.

I am trying to find the closed form of function c(p) satisfies following equations $$ c(p)=f(z,p)=\frac{-3p\log (z)}{p+(p-1)\frac{1}{z}-2(p-1)z+(\frac{1}{z})^{p-1}-2z^{p-1}} $$ and $$ c(p)=g(z,p)=\...
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Transcendental terms [closed]

What does it mean for two curves parametrised about 0 (say the graph of two functions of real variable) to 'differ only by transcendentally small terms'? How does this relates to their Taylor ...
xyz's user avatar
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Why can't $a=\theta-\sin(\theta)$ be solved for $\theta$ in terms of $a$ in closed form?

Context: I had taken an interest in alchemical symbols. Many of the ancient drawings are understandably crude, given the tools available at the time. In spite of their rough appearance, I imagined ...
Brovidio's user avatar
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How to prove a transcendental equation is never zero or sometimes zero?

I am an engineer by profession. Recently, while working on beams, I obtained the following transcendental equation: $$1-\cos(x)\cosh(x)+x^2\sin(x)\sinh(x)$$. I was wondering if this expression can ...
Thorongil's user avatar
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Approximate solutions to a transcendental equation of two variables

Say I have an equation of the form: $$ y = A + B\sin (x) +C\sin (x+Dy) $$ on the domain $0<x<2\pi$. I want to get $y$ as a function solely of $x$, i.e. remove the $y$-dependence from the R.H.S. ...
anonymous axolotl's user avatar
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Solve transcendental equation: At $α\ll 1$, $\left|\cos x + \alpha \frac{\sin x}{x}\right| > 1$, determine the width of of the $k$-th zone at $k\gg1$.

To solve this transcendental equations approximately: At $\alpha \ll 1$ find the positive solution of inequality: $\left|\cos x + \alpha \frac{\sin x}{x}\right| > 1$, they are divided into series ...
MICKEY's user avatar
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Solve transcendental equation : $\tanh(\alpha x) =\arctan(x)$, at $0<\alpha-1 \ll 1$ and at $\alpha\gg 1$?

To solve this transcendental equations approximately : Preivous: $\tanh(\alpha x) =\arctan(x)$, at $0<\alpha-1 \leq 1$ and at $\alpha\geq 1$. Edit: $\tanh(\alpha x) =\arctan(x)$, at $0<\alpha-1 ...
MICKEY's user avatar
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Complete factorisation of $x^8-2x^4\cos (4\theta)+1$ with roots of unity

Factorize completely $x^8-2x^4\cos (4\theta)+1$ using complex numbers and $n$-th root of unity. The answer given is $$\prod_{r=0}^3 \left(x^2-2x\cos\left(\theta+\frac{r\pi}{2}\right)+1\right)\,.$$ ...
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What is the currently accepted "correct" definition of a "transcendental function"?

Caveat: this question has already been asked on this site more than once, but the question has not been addressed completely. The question I want to ask is: there are two common definitions of a "...
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Help with volume integration application problem using Disk or Washer Methods, revolving about x-axis, revolving about y-axis.

I need to find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the given lines: y = $\sqrt {x}$ $y=0$, and $x=3$. A) the $x-axis$ B) the $y-...
jmp's user avatar
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Taking the inverse of a one-to-one polynomial

I'm trying to take the inverse of: $$f(x)=\frac{4x^3}{x^2+1}$$ When looking at the graph, it seems to be fully inversible (it is one-to-one), so I should be able to end up with another equation that ...
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Why do Fresnel-integrals contain $\sqrt{\pi}$?

The non-elementary functions $$ F(x) = \int \sin(x^2)\mathrm dx $$ $$ G(x) = \int \cos(x^2)\mathrm dx $$ will yield $$ F(x) =\sum_{k=1}^{\infty} (-1)^k \frac{x^{(4k+3)}}{(2k+1)!(4k+3)}$$ $$ G(x) =\...
Rickard Martensson's user avatar
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How to parameterize this particular symmetric ellipse related to the $\tanh(\ln(1+Z(t)^2)))$ with 4 points on the curve and 2 foci?

In this question, Ideas for parameterizing this curve in the complex plane and calculating its length by (numerical) contour integration?, I plotted the imaginary part of $$\tanh(\ln(1+Z(t)^2))$$ and ...
crow's user avatar
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Convex ratio of two Gamma functions

Is the function $\Gamma(x+1)^2/\Gamma(x+1/2)^2$ convex on $[0,+\infty[$ ?
Arno Berger's user avatar
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"Simple" condition that would prove a function transcendental

I conjectured that for every algebraic function $f(x)$ that is differentiable on $\mathbb{R}$, its $\lim_{x\to\infty}$ is either $\infty$, $-\infty$, or a finite value, so: If $f(x)$ is ...
Redbox's user avatar
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Integral $ \int_{-\infty}^{+\infty} e^{-t^2}\ln(1+e^t) \, dt $ [closed]

$$ \int_{-\infty}^{+\infty} e^{-t^2}\ln(1+e^t) \, dt $$ Hello everyone,i would like to know the result of the above integral and how to calculate or estimate it. background and progress so far:(1)...
proudmoore's user avatar
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Prove that $\int \sin(x^2)dx$ is not elementary

See edit It is known that the anti derivative of $\sin(x^2)$ is not an elementary function, and one can represent it using a power series by term-by-term integration of its Taylor series. However, is ...
user12986714's user avatar
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Are there simple conditions for $\frac{f(x)}{f'(x)}$ to be algebraic given $f(x)$ is not an algebraic function?

I should say in advance, I'm a physicist, so I apologise in advance for any mathematical faux pas! I've been attempting to prove some some properties of algebraic recurrence relations which converge ...
DoublyNegative's user avatar
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Possibility/impossibility of algebraic sum?

Suppose $ y = g(x_1) - g(x_2)$ on $[a,b],$ and there exists a smooth function $h$ over $[a,b] $ such that $h(y) = \frac{x_2}{x_1}.$ Then, does there exist a set of non-trivial functions $A$ algebraic ...
RandomWordMashup's user avatar
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1 answer
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Can a transcendental function generally be expressed as a polynomial of $n^{th}$ degree where $n$ approaches infinity?

A polynomial of $n^{th}$ degree can be expressed as a product of its roots: $$a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0=a_n(x-R_1)(x-R_2)...(x-R_n)$$ I'm wondering about a general rule for all functions, ...
MathAdam's user avatar
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Total derivative of transcendental function as a ratio of its partial derivatives

I have come across following: $$\frac{dI_{(V)}}{dV} = - \frac{\frac{\partial f_{(V,I)}}{\partial V}}{\frac{\partial f_{(V,I)}}{\partial I}}$$ where: $I_{(V)}$ is transcendental function (a function ...
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Prove that $e^{\frac{1}{\log(x)}}$ is at least countably transcendental

Q: Prove that $f(x)=e^{\frac{1}{\log(x)}}$ is at least countably transcendental for $x\in\Bbb R\cap (0,1).$
John Zimmerman's user avatar
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1 answer
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Closed-form solvability of elementary transcendental equations?

Fern-Ching Lin ([Lin 1983]) and Timothy Chow ([Chow 1999]) asked, when the solutions of a transcendental equation of elementary functions can be elementary numbers. My question is: To which more ...
Jürgen Will's user avatar
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How can you prove algebraic inputs of a transcendental function are transcendental?

In general, it's not true that a given transcendental function $f(z)$ will give you a transcendental output for countably infinite algebraic inputs, but is there a condition that can prove that ...
TeXnichal's user avatar
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What is transcendental equation/function?

I looked up several sources on the internet. A transcendental equation is an equation containing a transcendental function of the variable(s) being solved for. Such equations often do not have ...
iwbtrs's user avatar
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Does this transcendental equation have solutions for a non real variable?

Having solved the transcendental equation $e^{\frac{1}{\log(x)}}=x$ I found that it has solutions for a real variable $x.$ Does it have solutions for not real $x$ (i.e. over the complexes, ...
John Zimmerman's user avatar
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Show that the complex integral is a solution of Gauss's hypergeometric equation and express it in terms of standard ones.

There is Gauss's hypergeometric equation $[z(1-z)\frac{d^2}{dz^2} + [c - (a+b+1)z]\frac{d}{dz} - ab]u(z) = 0$ Apparently, (see [1], example in 14.2), this equation can be rearranged in the form $[(...
Lada Dudnikova's user avatar
2 votes
1 answer
778 views

Transcendental entire functions

Wikipedia defines a transcendental entire function as an entire function that is not polynomials. Is the following true? Given $f(z)$ an entire function, if there exists a polynomial $P \in \...
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Transcendence of $x^x = p \in \mathbb{P}$

I'm going straight for the question, so: let $x^x = p \in \mathbb{P}$, then $x$ is irrational. The proof is obvious, $p \neq a^a$ for some integer $a$, so $x^x = (\frac{a}{b})^{\frac{a}{b}}$, suppose ...
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The solutions for $x^x = z, z \in \mathbb{Z^+ - \{4\}}$ are irrational

Suppose that the function $x^x = z, z \in \mathbb{Z^+ - \{4\}}$ has rational solutions. Then $x^x = (\frac{a}{b})^{{\frac{a}{b}}}$ where $a$ and $b$ are coprimes. Now let's look at the equation $(\...
M.Silva's user avatar
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2 votes
2 answers
166 views

Are elementary compositions of nonelementary functions also nonelementary?

Say we have a nonelementary function $F(x)$ on the real numbers. Let $E_1,E_2,\ldots,E_n$ be a sequence of finite elementary functions on the reals. Is it always true that $$ R(x)=(E_1\circ E_2\circ \...
Lucas's user avatar
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2 answers
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Integer part of natural logarithm

Please, does anyone know of a algorithm to compute the integer part $n$ of natural logarithm of an integer $x$? $$n = \lfloor \ln(x) \rfloor$$ Preferably using integer arithmetic only (akin to ...
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Are polynomial functions with fractional exponents transcendental?

I'm having trouble categorizing fractional--order systems, that means functions like $$ f(x) = K \cdot (1 + \sqrt{x})$$ or more generally, e.g. $$ g(x) = \dfrac{a_0 + a_1 x^{1/2} + a_2 x^{1} + a_3 ...
Robert Seifert's user avatar