Questions tagged [transcendental-functions]

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

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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 ...
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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 ...
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Transcendence of limit function under uniform convergence.

Suppose $\{f_n\}$ is a sequence of non-constant meromorphic functions defined on some domain $D\subset\mathbb{C}.$ Assume that $\{f_n\}$ converges uniformly (w.r.t spherical metric) on $D$ to some non-...
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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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20 views

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-...
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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) =\...
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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 ...
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Convex ratio of two Gamma functions

Is the function $\Gamma(x+1)^2/\Gamma(x+1/2)^2$ convex on $[0,+\infty[$ ?
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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 ...
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108 views

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)...
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transcendental function's oblique asymptotes

It is known that function like $\,f(x)=x+\sin(1/x)\,$ has oblique aymptote, which is $y=x.$ However, if $\,P(x)\,$ and $\,Q(x)\,$ is composed of only transcendental function, let $\,g(x)=P(x)/Q(x).$ ...
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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 ...
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31 views

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 ...
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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 ...
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26 views

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, ...
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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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1answer
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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 $x\in\Bbb R\cap (0,1).$
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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 ...
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44 views

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 ...
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706 views

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 ...
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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, ...
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54 views

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 $[(...
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1answer
252 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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54 views

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 $(\...
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136 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 \...
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558 views

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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372 views

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 ...
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1answer
127 views

Evaluation of :$\sum_{n=1}^{\infty}\frac{\sin n \log n}{n}$

I have tried to evaluate this sum $\sum_{n=1}^{\infty}\frac{\sin n \log n}{n}$ using :$\sum_{n=1}^{\infty}\frac{\sin n }{n}$ , Really the partial sum of the titled series given by polylogarithm ...
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2answers
108 views

Transcendental numbers

Can transcendental numbers be plotted? Also, can a computer recognize a transcendental number? I mean, for example, a computer, while computing, understand that the number it is computing is not ...
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Multi-logarithm generalisation with multipliers

I previously mentioned a proposed "multi-stage logarithm" function, and managed to come up with a generalisation of the function. Originally, the multi-logarithm was defined as: $a_0^x+a_1^x+...+...
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how to solve $A\ln(1+t)-A\frac{t}{1+t} + B\frac{\ln^2(1+t)}{t^2}=0$

Does anyone have any ideas on how to solve $t$ on $A\ln(1+t)-A\frac{t}{1+t} + B\frac{\ln^2(1+t)}{t^2}=0$, where $A$ and $B$ are constants. It seems impossible to solve such an complex function. Is ...
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how to solve a $\log(1+\frac1x)*x$ function

I know lambert function is available to solve function like $xln(x)$, I wonder if there is a similar way I can solve function $b - x log_2(1+\frac{a}{x})=0$.
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1answer
215 views

Transcendental and implicit functions

Def. An implicit function is a function that is defined implicitly by an implicit equation, by associating one of the variables (the value) with the others (the arguments). For example, $$F(x,y)=0$$$$...
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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 ...
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68 views

How to calculate $\int_{ }^{ }\frac{14x+2}{x^2+4}dx$

This problem was given to me as part of a transcendental functions homework: $$\int_{ }^{ }\frac{14x+2}{x^2+4}dx$$ I believe this problem requires substitution as well as sec-1 but i'm having ...
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1answer
62 views

Laplace transform of $t^\mu I_\nu(at)$

According to Vol. I of Bateman (p. 196 in my copy), $$ \int\limits_0^{+\infty} t^\mu I_\nu(at)\,e^{-pt} dt = \Gamma(\mu+\nu+1) \, s^{-\mu-1} \, P_\mu ^{-\nu}(p/s), $$ where $P_\mu^{-\nu}$ is the ...
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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 ...
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Any theorems on the derivatives of transcendental functions?

I am interested if there are any theorems/conjectures on the rates of change of transcendental functions. More specifically, I want to know that if $f(x)$ is a transcendental function, and $f(a)\in \...
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112 views

Is sinx/x a transcendental function? [closed]

Or is it algebraic? Is there any properties of transcendental functions that allow me to recognize them?
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1answer
636 views

What are the types of transcendental functions?

I was reading through my Calculus Textbook. When I got the chapter on transcendental functions it mentioned how to integrate and differentiate them. The only functions it mentioned were exponential, ...
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43 views

How to prove that a logarithmic integral is transcendental?

After I've read, from [1], a proof that the logarithm $\log (x)$, defined for $x>0$, is a trancendental function, I wondered what should be the argument to prove (I believe that it holds) that the ...
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1answer
83 views

Why my calculator is showing a weird result.

I was trying to solve the equation $x^{\pi}-\pi^x=0$ using numerical analysis(Using Bisection method ,Regula Falsi method). I thought $0$ would be a good start. So I plugged $0^{\pi} $ and it showed ...
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1answer
263 views

Could someone explain this proof - any non-constant element of $F(x)$ transcendental over $F$

In a paper, I found a proof that any is non-constant element of the rational functions $F(x)$ over the field $F$ is transcendental with respect to the base field $F$. However, there is something in ...
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152 views

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 $q_2=\dfrac{1}{2}$?
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1answer
56 views

Does the derivative of a holomorphic function $f(t)$ always vanish at multiple roots of $f$?

Does the derivative of a holomorphic function $f(t)$ always vanish at multiple roots of $f$? I know its true for polynomials, but is it a general fact known to hold for all transcendental functions?
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2answers
352 views

All roots to one variable transcendental equation

How to find all possible roots for one variable transcendental equation? The equation is as follows: $\cosh(x)\cos(x)+1=0$ How should I solve the above equation if it has two variables? $\cosh(x)\cos(...
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132 views

Solution for two variable transcendental equation

How to find the solution for a two variable transcendental equation? The following is the equation: $$a^5+ab^4+2a^3b^2\cos(100a)\cosh(100b)+(a^4b-a^2b^3)\sin(100a)\sinh(100b)=0$$ I know that there ...