Mathematics Stack Exchange
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Questions tagged [transcendental-functions]

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

A function $f(x)$ is transcendental if there it does not satisfy an algebraic equation. These extend the notion of transcendental (and algebraic) numbers. Examples include $e^x,\sin(x),\log(x)$; non-examples include polynomials, radicals, rational functions, and characteristic functions; note that non-transcendental (i.e., algebraic) functions need not be elementary.

This tag should often be used for questions asking whether a function is transcendental. In particular, the indefinite integral of an algebraic function, such as $\int 1/x \,dx$, is often transcendental.

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In simple English, what does it mean to be transcendental?

From Wikipedia A transcendental number is a real or complex number that is not algebraic A transcendental function is an analytic function that does not satisfy a polynomial equation However these definitions are arguably rather cryptic to those…
AlanSTACK
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Is $\Phi(q)$ rational for some $q \in \mathbb{Q}^*$, where $\Phi$ is the standard normal cumulative distribution function?

Suppose that we have rational numbers $q_1$, $q_2$ such that $$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{q_1}e^{-\frac{t^2}{2}} \,\mathrm{d}t=q_2.$$ Does this imply that $q_1=0$ and $q_2=\dfrac{1}{2}$?
Jaakko Seppälä
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Is there an entire function with $f(\mathbb{Q}) \subset \mathbb{Q}$ and a non-finite power series representation having only rational Coeffitients

I'm trying to answer the following question: Is there an entire function $f(z) := \sum \limits_{n=0}^\infty c_nz^n$ such that $f(\mathbb{Q}) \subset \mathbb{Q}$ $\forall n: c_n \in \mathbb{Q}$ $f$ is not a polynomial ? I'm trying to show that no…
Felicitas
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Why can't $y=xe^x$ be solved for $x$?

I apologize for my mathematical ignorance regarding this, but could someone help me understand why it isn't possible to (symbolically) find an inverse function for $f(x)=xe^x$? The most obvious (but presumably the most trivial) is that $f$ does not…
jacobq
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How did Leibniz prove that $\sin (x)$ is not an algebraic function of $x$?

In the Wikipedia article about transcendental numbers we can read the following: The name "transcendental" comes from Leibniz in his 1682 paper where he proved that sin(x) is not an algebraic function of x. I would like to know can someone…
Farewell
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Simple recursive algorithms to manually compute elementary functions with pocket calculators

Let $x_n\,(n\in\Bbb N)$ be the sequence defined by $$x_{n+1}=\frac{x_n}{\sqrt{x_n^2+1}+1}\tag 1$$ then it's well know that $2^nx_n\xrightarrow{n\to\infty}\arctan(x_0)$. This gives a very simple recursive algorithm to manually compute $\arctan$ on a…
Fabio Lucchini
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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 "transcendental function", both of which are readily…
Prime Mover
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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 there any way to show that it's antiderivative is…
user12986714
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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 differentiable everywhere on $\mathbb{R}$ and its…
Mr Pink
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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, quaternions, octonions)? Edit, June 24th 2020: I plotted…
geocalc33
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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 is mirrored in the $x=y$ axis. However, I cannot…
Flame
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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 conjectured in The Hyperbolic Tangent of the…
crow
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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 general kinds of transcendental equations can Lin's…
Jürgen Will
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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 closed-form solutions. And then transcendental…
Ilya Stokolos
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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 $P \in \mathbb{C}[X,Y]$ with the property that $P(z,f(z)) = 0$…
JonHales
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