Questions tagged [transcendental-functions]
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questions
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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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1answer
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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2answers
65 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 quaternions, octonions)...
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49 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
$[(...
1
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1answer
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 ...
1
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2answers
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 $(\...
2
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2answers
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
$$ R(x)=(E_1\circ E_2\circ \...
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2answers
221 views
Integer part of natural logarithm
Please, does anyone know of a algorithm to compute the integer part $n$ of natural logarithm of an integer $x$?
$$n = \lfloor \ln(x) \rfloor$$
Preferably using integer arithmetic only (akin to ...
0
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2answers
98 views
Are polynomial functions with fractional exponents transcendental?
I'm having trouble categorizing fractional--order systems, that means functions like
$$ 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 ...
2
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1answer
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 ...
1
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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 ...
1
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0answers
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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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 ...
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1answer
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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 $b - x log_2(1+\frac{a}{x})=0$.
1
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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,
$$F(x,y)=0$$$$...
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4answers
606 views
Why can't $y=xe^x$ be solved for $x$?
I apologize for my mathematical ignorance regarding this, but could someone help me understand why it isn't possible to (symbolically) find an inverse function for $f(x)=xe^x$?
The most obvious (but ...
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2answers
55 views
How to calculate $\int_{ }^{ }\frac{14x+2}{x^2+4}dx$
This problem was given to me as part of a transcendental functions homework:
$$\int_{ }^{ }\frac{14x+2}{x^2+4}dx$$
I believe this problem requires substitution as well as sec-1 but i'm having ...
1
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1answer
58 views
Laplace transform of $t^\mu I_\nu(at)$
According to Vol. I of Bateman (p. 196 in my copy),
$$
\int\limits_0^{+\infty} t^\mu I_\nu(at)\,e^{-pt} dt = \Gamma(\mu+\nu+1) \, s^{-\mu-1} \, P_\mu ^{-\nu}(p/s),
$$
where $P_\mu^{-\nu}$ is the ...
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0answers
155 views
Simple recursive algorithms to manually compute elementary functions with pocket calculators
Let $x_n\,(n\in\Bbb N)$ be the sequence defined by
$$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 ...
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0answers
46 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 \...
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2answers
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?
0
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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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0answers
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 ...
2
votes
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 ...
2
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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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0answers
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.$$
Does this imply that $q_1=0$ and $q_2=\dfrac{1}{2}$?
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1answer
48 views
Does the derivative of a holomorphic function $f(t)$ always vanish at multiple roots of $f$?
Does the derivative of a holomorphic function $f(t)$ always vanish at multiple roots of $f$?
I know its true for polynomials, but is it a general fact known to hold for all transcendental functions?
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2answers
269 views
All roots to one variable transcendental equation
How to find all possible roots for one variable transcendental equation?
The equation is as follows:
$\cosh(x)\cos(x)+1=0$
How should I solve the above equation if it has two variables?
$\cosh(x)\cos(...
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0answers
113 views
Solution for two variable transcendental equation
How to find the solution for a two variable transcendental equation?
The following is the equation:
$$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 ...
2
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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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1answer
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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1answer
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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2answers
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$ ...
2
votes
2answers
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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0answers
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 ...
0
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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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11answers
18k views
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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2answers
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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$$
from elementary calculus. And at some other place (or maybe the same book even) I've learned that
$$\...
