Mathematics Stack Exchange
Mathematics Stack Exchange

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 in math?

From Wikipedia, we have the following definitions: 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…
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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Inverse function of $y=\frac{\ln(x+1)}{\ln x}$

I've been wondering for a while if it's possible to find the inverse function of $y=\frac{\ln(x+1)}{\ln x}$ over the reals. This is the same as finding the positive real root of $x^y-x-1$. I realize that it's impossible with elementary functions,…
B H
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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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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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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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Seeking clarification: Convergence of the series $\sum_{n=1}^{\infty} \frac{1}{e^{n^2}}$

I was asked to show whether the following series converges: $$\sum_{n=1}^{\infty}\frac{1}{{e^{n^2}}}$$ Here's my first attempt which I thought was pretty straightforward: $$e^{n^{2}}=(e^{n})^{n}\ge e^{n} \ ,\ \forall \ n \in \Bbb{N}$$ $$\implies…
bwootton
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Inverse of $f(x)=\sin(x)+x$

What is the inverse of $$f(x)=\sin(x)+x.$$ I thought about it for a while but I couldn't figure it out and I couldn't find the answer on the internet. What about $$f(x)=\sin(a \cdot x)+x$$ where $a$ is a known real constant. Thank you for taking…
Mircea
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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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Solution of a transcendental equation

Let $a_1,a_2,a_3,a_4,a_5 \in \mathbb{R}$, $a_1 \neq 0$ and consider the transcendental equation $$ (1) \quad a_1x^2 + a_2x + a_3 + a_4 e^{-2a_5x} + a_6 xe^{-a_5x} + a_7e^{-a_5x} = 0. $$ Is there a way to solve explicitly this equation? I know how…
ThiagoGM
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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 following: $$\cos(y)=\frac{x-1}{2\sqrt{x}}$$ If I then…
lynx_s
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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 to find $x$ as a function of $y$ and don't know…
thelordabdo
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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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How to invert this function?

I need to invert this function: $$ y=\frac{\ln(x)}{\ln(x-1)}+1 $$ The domain is real (for x>1 and x!=2) Why can't we just divide it like this: $$ y=ln(x-(x-1))+1 $$ and then it's: $$ y=ln(1)+1 $$ so it seems wrong. Where did I make the mistake?
TomDavies92
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