Questions tagged [transcendental-functions]
Transcendental functions are those functions that do not satisfy an algebraic equation.
60
questions
1
vote
1answer
65 views
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 ...
0
votes
2answers
43 views
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 ...
0
votes
0answers
27 views
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-...
1
vote
2answers
54 views
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
$$\prod_{r=0}^3 \left(x^2-2x\cos\left(\theta+\frac{r\pi}{2}\right)+1\right)\,.$$
...
4
votes
1answer
98 views
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 "...
0
votes
1answer
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-...
2
votes
2answers
51 views
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 ...
0
votes
0answers
75 views
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
$$ F(x) =\sum_{k=1}^{\infty} (-1)^k \frac{x^{(4k+3)}}{(2k+1)!(4k+3)}$$
$$ G(x) =\...
2
votes
0answers
39 views
How to parameterize this particular symmetric ellipse related to the $\tanh(\ln(1+Z(t)^2)))$ with 4 points on the curve and 2 foci?
In this question, Ideas for parameterizing this curve in the complex plane and calculating its length by (numerical) contour integration?, I plotted the imaginary part of $$\tanh(\ln(1+Z(t)^2))$$ and ...
0
votes
0answers
25 views
Convex ratio of two Gamma functions
Is the function $\Gamma(x+1)^2/\Gamma(x+1/2)^2$ convex on $[0,+\infty[$ ?
3
votes
0answers
89 views
“Simple” condition that would prove a function transcendental
I conjectured that for every algebraic function $f(x)$ that is differentiable on $\mathbb{R}$, its $\lim_{x\to\infty}$ is either $\infty$, $-\infty$, or a finite value, so:
If $f(x)$ is ...
0
votes
1answer
108 views
Integral $ \int_{-\infty}^{+\infty} e^{-t^2}\ln(1+e^t) \, dt $ [closed]
$$ \int_{-\infty}^{+\infty} e^{-t^2}\ln(1+e^t) \, dt $$
Hello everyone,i would like to know the result of the above integral and how to calculate or estimate it.
background and progress so far:(1)...
0
votes
1answer
24 views
transcendental function's oblique asymptotes
It is known that function like $\,f(x)=x+\sin(1/x)\,$ has oblique aymptote, which is $y=x.$
However, if $\,P(x)\,$ and $\,Q(x)\,$ is composed of only transcendental function, let $\,g(x)=P(x)/Q(x).$ ...
4
votes
0answers
155 views
Prove that $\int \sin(x^2)dx$ is not elementary
See edit
It is known that the anti derivative of $\sin(x^2)$ is not an elementary function, and one can represent it using a power series by term-by-term integration of its Taylor series.
However, is ...
0
votes
1answer
31 views
Are there simple conditions for $\frac{f(x)}{f'(x)}$ to be algebraic given $f(x)$ is not an algebraic function?
I should say in advance, I'm a physicist, so I apologise in advance for any mathematical faux pas!
I've been attempting to prove some some properties of algebraic recurrence relations which converge ...
0
votes
0answers
41 views
Possibility/impossibility of algebraic sum?
Suppose $ y = g(x_1) - g(x_2)$ on $[a,b],$ and there exists a smooth function $h$ over $[a,b] $ such that $h(y) = \frac{x_2}{x_1}.$
Then, does there exist a set of non-trivial functions $A$ algebraic ...
1
vote
1answer
26 views
Can a transcendental function generally be expressed as a polynomial of $n^{th}$ degree where $n$ approaches infinity?
A polynomial of $n^{th}$ degree can be expressed as a product of its roots:
$$a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0=a_n(x-R_1)(x-R_2)...(x-R_n)$$
I'm wondering about a general rule for all functions, ...
1
vote
0answers
24 views
Total derivative of transcendental function as a ratio of its partial derivatives
I have come across following:
$$\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 ...
1
vote
1answer
39 views
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).$
2
votes
1answer
117 views
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 ...
0
votes
0answers
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 ...
1
vote
1answer
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 ...
3
votes
3answers
111 views
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, ...
0
votes
0answers
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
$[(...
2
votes
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
\...
0
votes
0answers
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 ...
1
vote
2answers
44 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 $(\...
2
votes
2answers
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 \...
0
votes
2answers
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 ...
0
votes
2answers
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.
$$ g(x) = \dfrac{a_0 + a_1 x^{1/2} + a_2 x^{1} + a_3 ...
2
votes
1answer
127 views
Evaluation of :$\sum_{n=1}^{\infty}\frac{\sin n \log n}{n}$
I have tried to evaluate this sum $\sum_{n=1}^{\infty}\frac{\sin n \log n}{n}$ using :$\sum_{n=1}^{\infty}\frac{\sin n }{n}$ , Really the partial sum of the titled series given by polylogarithm ...
1
vote
2answers
108 views
Transcendental numbers
Can transcendental numbers be plotted? Also, can a computer recognize a transcendental number? I mean, for example, a computer, while computing, understand that the number it is computing is not ...
1
vote
0answers
28 views
Multi-logarithm generalisation with multipliers
I previously mentioned
a proposed "multi-stage logarithm" function, and managed to come up with a generalisation of the function.
Originally, the multi-logarithm was defined as:
$a_0^x+a_1^x+...+...
0
votes
1answer
44 views
how to solve $A\ln(1+t)-A\frac{t}{1+t} + B\frac{\ln^2(1+t)}{t^2}=0$
Does anyone have any ideas on how to solve $t$ on $A\ln(1+t)-A\frac{t}{1+t} + B\frac{\ln^2(1+t)}{t^2}=0$, where $A$ and $B$ are constants. It seems impossible to solve such an complex function. Is ...
-3
votes
1answer
31 views
how to solve a $\log(1+\frac1x)*x$ function
I know lambert function is available to solve function like $xln(x)$, I wonder if there is a similar way I can solve function $b - x log_2(1+\frac{a}{x})=0$.
1
vote
1answer
215 views
Transcendental and implicit functions
Def. An implicit function is a function that is defined implicitly by an implicit equation, by associating one of the variables (the value) with the others (the arguments). For example,
$$F(x,y)=0$$$$...
9
votes
5answers
2k views
Why can't $y=xe^x$ be solved for $x$?
I apologize for my mathematical ignorance regarding this, but could someone help me understand why it isn't possible to (symbolically) find an inverse function for $f(x)=xe^x$?
The most obvious (but ...
0
votes
2answers
68 views
How to calculate $\int_{ }^{ }\frac{14x+2}{x^2+4}dx$
This problem was given to me as part of a transcendental functions homework:
$$\int_{ }^{ }\frac{14x+2}{x^2+4}dx$$
I believe this problem requires substitution as well as sec-1 but i'm having ...
1
vote
1answer
62 views
Laplace transform of $t^\mu I_\nu(at)$
According to Vol. I of Bateman (p. 196 in my copy),
$$
\int\limits_0^{+\infty} t^\mu I_\nu(at)\,e^{-pt} dt = \Gamma(\mu+\nu+1) \, s^{-\mu-1} \, P_\mu ^{-\nu}(p/s),
$$
where $P_\mu^{-\nu}$ is the ...
6
votes
0answers
197 views
Simple recursive algorithms to manually compute elementary functions with pocket calculators
Let $x_n\,(n\in\Bbb N)$ be the sequence defined by
$$x_{n+1}=\frac{x_n}{\sqrt{x_n^2+1}+1}\tag 1$$
then it's well know that $2^nx_n\xrightarrow{n\to\infty}\arctan(x_0)$.
This gives a very simple ...
1
vote
0answers
50 views
Any theorems on the derivatives of transcendental functions?
I am interested if there are any theorems/conjectures on the rates of change of transcendental functions. More specifically, I want to know that if $f(x)$ is a transcendental function, and $f(a)\in \...
-1
votes
2answers
112 views
Is sinx/x a transcendental function? [closed]
Or is it algebraic? Is there any properties of transcendental functions
that allow me to recognize them?
0
votes
1answer
636 views
What are the types of transcendental functions?
I was reading through my Calculus Textbook. When I got the chapter on transcendental functions it mentioned how to integrate and differentiate them. The only functions it mentioned were exponential, ...
2
votes
0answers
43 views
How to prove that a logarithmic integral is transcendental?
After I've read, from [1], a proof that the logarithm $\log (x)$, defined for $x>0$, is a trancendental function, I wondered what should be the argument to prove (I believe that it holds) that the ...
2
votes
1answer
83 views
Why my calculator is showing a weird result.
I was trying to solve the equation $x^{\pi}-\pi^x=0$ using numerical analysis(Using Bisection method ,Regula Falsi method). I thought $0$ would be a good start. So I plugged $0^{\pi} $ and it showed ...
2
votes
1answer
263 views
Could someone explain this proof - any non-constant element of $F(x)$ transcendental over $F$
In a paper, I found a proof that any is non-constant element of the rational functions $F(x)$ over the field $F$ is transcendental with respect to the base field $F$. However, there is something in ...
14
votes
0answers
152 views
Is $\Phi(q)$ rational for some $q \in \mathbb{Q}^*$, where $\Phi$ is the standard normal cumulative distribution function?
Suppose that we have rational numbers $q_1$, $q_2$ such that
$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{q_1}e^{-\frac{t^2}{2}} \,\mathrm{d}t=q_2.$$
Does this imply that $q_1=0$ and $q_2=\dfrac{1}{2}$?
0
votes
1answer
56 views
Does the derivative of a holomorphic function $f(t)$ always vanish at multiple roots of $f$?
Does the derivative of a holomorphic function $f(t)$ always vanish at multiple roots of $f$?
I know its true for polynomials, but is it a general fact known to hold for all transcendental functions?
0
votes
2answers
352 views
All roots to one variable transcendental equation
How to find all possible roots for one variable transcendental equation?
The equation is as follows:
$\cosh(x)\cos(x)+1=0$
How should I solve the above equation if it has two variables?
$\cosh(x)\cos(...
0
votes
0answers
132 views
Solution for two variable transcendental equation
How to find the solution for a two variable transcendental equation?
The following is the equation:
$$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 ...
