Questions tagged [transcendental-functions]

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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 ...
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60 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 quaternions, octonions)...
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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 $[(...
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65 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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50 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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41 views

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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120 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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221 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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98 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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97 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
79 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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20 views

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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44 views

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
130 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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606 views

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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55 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
58 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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55 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
434 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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38 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
80 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
173 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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132 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
48 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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269 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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113 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 ...
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1answer
60 views

What functions do we need besides polynomials to describe any real as the root of some equation?

If $f(x)$ is some polynomial with integer coefficient of degree $>0$, then any solution to $f(x) = 0$ is an algebraic number. If $g(x)$ and $h(x)$ are also polynomials with integer coefficients, is ...
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84 views

How can I check the parity of transcendental functions?

I know how to check it in general ($f(x)=f(-x)$) but I don't understand how I can check it for any transcendental functions, because I cannot check if (for example) $\tan(x)= \tan(-x)$
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68 views

On the behavior of transcendental functions

Given that single-variable algebraic functions take on algebraic values when the input is algebraic, and take on transcendental values when the input is transcendental, and knowing that transcendental ...
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179 views

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 $f(\mathbb{Q}) \subset \mathbb{Q}$ $\forall n: c_n \in \mathbb{Q}$ $f$ ...
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125 views

Why is $e^g$ transcendental over $\mathbb{C}(z)$ for $g$ rational?

In the 1972 paper Integration in Finite Terms by M Rosenlicht he says: we note that if $g(z)$ is is a non-constant rational function of the complex variable $z$ then $e^g$ is not algebraic over $\...
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52 views

Derivatives bounded by a constant

I want to find a function such that $f(x)$ is bounded by a constant $c$, i.e., $f(x)<c$. The derivative of the function must also be bounded by a constant, i.e., $f'(x)<c$ but the second ...
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1answer
66 views

Can transcendental functions be approximated to algebraic ones.

I just wanted sin(x) in terms of algebraic operations like addition,multiplication,etc. And can we extract that from the right angled triangle sides ratio definition, sine=altitude/hypotenuse or we ...
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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 ...
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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 ...
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Which functions can be constructed as a limit of monomials of a polynomial?

Most of us know the famous limit $$\lim_{n \rightarrow \infty}\left(1+\frac1n\right)^n = e$$ from elementary calculus. And at some other place (or maybe the same book even) I've learned that $$\...