For question involving exponential functions and questions on exponential growth or decay.

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9
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31 views

$\int_{- \infty}^{\infty} \frac{f(x)}{1+\exp{g(x)}}dx=\int_{0}^{\infty} f(x) dx$ for $f(x)=f(-x),~g(x)=-g(-x)$ - are there other formulas like that?

If $f(x)$ any even function, integrable on $(0,\infty)$ and $g(x)$ any odd function, then we have: $$\int_{- \infty}^{\infty} \frac{f(x)}{1+e^{g(x)}}dx=\int_{0}^{\infty} f(x) dx \tag{1}$$ The ...
9
votes
0answers
72 views

$\pi$ base $e$ or $\pi=\sum\limits_{n=-1}^{\infty} a_ne^{-n}$ where $a_n\in\{0,1,-1\}$

I was "playing with $\pi$" trying to look at it in different numeral systems and it's not so hard to obtain $\pi$ base $2$ or $3$ or even $\varphi=\frac{\sqrt{5}+1}{2}$, using Maclaurin series of $\...
6
votes
0answers
109 views

An integral for $2\pi+e-9$

Motivation Lucian asked about the almost-integer $2\pi+e\approx9$ in a comment to a partially answered why question about $e\approx H_8$. This is more involved than approximations to $\pi$ and ...
6
votes
0answers
259 views

Rational series representation of $e^\pi$

This question is related to Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$? by Tito Piezas III. Andrew Fraker (2014) found an almost-integer which is equivalent to the following ...
6
votes
0answers
197 views

Closed-form of integrals containing double exponentials

Are there closed forms for the following integrals? $$\begin{align} I_1(w) & = \int_{-\infty}^{\infty} \frac{\exp(-we^y)}{y^2+\pi^2} dy, \\ I_2(w) & = \int_{-\infty}^{\infty} \frac{\exp(-we^y+...
5
votes
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63 views

Generalized Trigonometric Functions in terms of exponentials and roots of unity

I am trying to come up with generalized trigonometric functions using the exponential definition that we use today for the trig functions sine and cosine $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}; \cos x =\...
5
votes
0answers
76 views

Integral, possibly of Bessel or Exponential form.

I'm working with a hierarchical statistical model, whereby the output of a log-normal distribution affects the argument of an exponential distribution. I need to marginalize, obtaining the following ...
4
votes
0answers
190 views

Is Every transcendental entire function $f(z) = C + \exp a + \exp b + \exp c $?

Let $z$ be a complex number and let $f(z)$ be any transcendental entire function. Is it true that $f(z) = C + \exp a + \exp b + \exp c $ where $ a,b,c $ are entire functions of $z$ and $C$ is a ...
4
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0answers
80 views

Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum $$\...
4
votes
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56 views

How quickly can one compare exp(m/n) to a given rational?

For positive integers $\hspace{.06 in}m_{\hspace{.02 in}0}\hspace{.02 in},n_0\hspace{.02 in},m_1,n_1\:$, $\;$ how difficult is it to decide whether $$\exp\left(\hspace{-0.03 in}\frac{m_{\hspace{.02 in}...
3
votes
0answers
67 views

Does this equation have no solutions?

The question is this : The source from where I got this question was devoid of any answers to it, so I came here, this is how I proceeded : LHS : $((((({(x)^x})^{2x})^{3x})^{....x^2})^2 = (((((x)^...
3
votes
0answers
108 views

Solving equation involving factorials

I have this particular equation $\frac{(\alpha-1)!(\beta-1)!}{(\alpha+\beta-1)!} = \frac{\Gamma(p)(1+q)^{n+2p} 2^n}{q^{p}(2+q)^{n+p}}$. Now, given the values of $\alpha$ and $\beta$, I need to find ...
3
votes
0answers
49 views

Proof that $|1 - e^{i \theta}| \geq \frac{2|\theta|}{\pi}$ for $-\pi \leq \theta \leq \pi$?

I would like to prove (geometrically if possible) the above result. Could someone help?
3
votes
0answers
63 views

Invertibility of infinite order matrix

how the matrix $[e^{-(x_j-x_k)^2}]$ is invertible where $\{x_j\}$ be any real sequence such that $(x_{j+1}-x_j) >0$ for all $j \in Z$ where $Z$ denotes the set of integers.
3
votes
0answers
59 views

What is $\int \frac{e^{a x}}{1+x^2} dx $?

In my answer to another question (here: Upper and lower bound on different of ${\rm erf}(\frac{x+c}{b})-{\rm erf}(\frac{x-c}{b})$), I came up with this integral: $\int \frac{e^{a x}}{1+x^2} dx $. I ...
3
votes
0answers
55 views

Interpreting and understanding the identity $e^{iz} = \cos(z) \pm \sqrt{\cos^2(z) - 1}$

A question in my complex analysis book (Gamelin's "Complex Analysis", question I.8.7) asks me to prove that $e^{iz} = \cos(z) \pm \sqrt{\cos^2(z) - 1}$. Using the identity $\cos(z) = \frac{e^{iz} + e^{...
3
votes
0answers
109 views

Is there a way to exploit local redundancy in a function to speed up Monte Carlo integration?

In every Monte Carlo method I've ever seen, $f$ must be recomputed from scratch for each point that is (somehow randomly) selected to contribute to the overall integral. However, most functions have ...
3
votes
0answers
220 views

Solving an exponential equation without the quadratic formula

High school math student here. In my homework I was asked to solve $16^x +4^{x+1} - 3= 0$ and I used substitution to get $x=\log_4{(-2+\sqrt7)}$. However, this was in the chapter on ...
3
votes
0answers
75 views

How to find the Maclaurin series for the integral of $e^{x^2}$?

I am trying to find the Maclaurin series for the integral of $e^{x^2}$? What I done so far is that the Maclaurin series for $e^{x^2}$ is $$e^{x^2}=\sum_{n=0}^{\infty}\frac{x^{2n}}{n!}$$ So would ...
3
votes
0answers
70 views

Why is the base of an exponential function limited to the set of real numbers greater than zero?

From Rogawski, Jon (2011-04-01). Single Variable Calculus (Page 339). W. H. Freeman. An exponential function is a function of the form $f (x) = b^x$, where $b > 0$ and $b \neq 1$. Why is $b$ ...
3
votes
0answers
74 views

Roots of derivative of q-expontial function

Let the q-deformation of the exponential function be defined by $$ e_q(z)=\sum_{n=0}^\infty{\frac{z^n}{[n]_q!}}. $$ Eq. (1.8) of this paper provides the product representation $$ e_q(z)=\prod_{k=0}^...
3
votes
0answers
74 views

Showing the exponential and logarithmic functions are unique in satisfying their properties

The question asks to prove that there exists a unique function defined on $\Bbb R$ and satisfying the following conditions: 1) $f(1) = a$ $(a>0, a \neq 0)$ 2) $f(x_1) \cdot f(x_2) = f(x_1 + x_2)$...
3
votes
0answers
69 views

How general is the convergence of the exponential function's power series?

Let $\mathbf{V}$ be a Fréchet space whose underlying set is $V$. Let $\;\; \beta \: : \: V\times V \: \to \: V \;\;$ be a continuous bilinear map that has an identity element and is power-...
3
votes
0answers
638 views

How to prove $x^y$ is jointly continuous?

It's known that real exponentiation $x^y$ is continuous in each variable, but is real exponentiation jointly continuous in both the exponent and the base? I considering the function $(0,\infty)\times\...
3
votes
0answers
56 views

How to solve the equation $ (x-2)^{\log_{100}(x-2)}+\log_{10}(x-2)^5-12 = 10^{2\log_{10}(x-2)}$?

If $\displaystyle (x-2)^{\log_{10^2}(x-2)}+\log_{10}(x-2)^5-12 = 10^{2.\log_{10}(x-2)}$, then value of $x$ is ... My try:: Let $\log_{10}(x-2) = y\Leftrightarrow (x-2)=10^y$. Then $(10)^{y.\frac{1}{...
3
votes
0answers
117 views

Prove (*) by induction on k.

Challenge: For linear systems with constant coefficients, in some sense we "never need more" than the so-called exponential polynomials, meaning expressions in the form $$\sum_{i=1}^m c_it^{n_i}e^{\...
3
votes
0answers
252 views

Infinite series of matrices almost but not quite matrix exponential

I'm working on a problem that has brought up for me the need to address infinite series of the following form, $$ \sum_{i=k}^\infty \frac{1}{i!}A^{i-k+1} $$ where $A$ is an $n\times n$ matrix. If $k = ...
2
votes
0answers
30 views

Analytical integration of product of exponential functions

I am trying to obtain an analytical formula for the following integral. My first question is whether it is possible to obtain an analytical formula without the use of transcendental functions. My ...
2
votes
0answers
21 views

Integral of third order polynomial exponential

I am looking for approximated or exact solution of \begin{align} I = \int_R \exp(cx^3-ax^2+bx)dx \end{align} where $a,b,c$ are complex numbers defined as: \begin{align} c &= \frac{1}{3}i\pi\phi'''...
2
votes
0answers
41 views

Estimating $n!$ as $e \left(\frac ne \right)^n \le n! \le ne \left(\frac ne \right)^n$

I'm told that for $n \geq 2,$ $$\sum_{k=1}^{n-1} f(k) \leq \int_1^n f(x) \, dx \leq \sum_{k=2}^n f(k)$$ I am then asked to consider $\ln n! = \sum_{k=1}^n \ln k$ and show that for $n \geq 2$ $$n! \...
2
votes
0answers
53 views

Is there a way to solve the exponential equation $a^x + b^x + c^x = d$ analytically?

So I came across this equation. $$a^x + b^x + c^x = d$$ where $a, b, c$ and $d$ are all constants. And I just wondered, is there any way to solve for x analytically?
2
votes
0answers
56 views

Does the continued fraction for $e^{3/n}$ have a pattern?

Wikipedia has patterns for the simple continued fractions $e^{1/n},e^{2/n}$, which made me wonder whether there is one known for $e^{3/n}?$ (by pattern, I mean that the partial quotients $a_n$ can ...
2
votes
0answers
32 views

Baker Campbell Hausdorff formula and bernoulli numbers

The BCH formula states that the product of two exponentials of non commuting operators can be combined into a single exponential involving commutators of these operators. One may write that $\ln(e^A e^...
2
votes
0answers
43 views

The number $e$ approximated as sums of of $X \sim U(0,1)$. Why does it work?

In this post a computer simulation to approximate $e$ is based on the mathematical knowledge that $E[\xi]=e$, where $\xi$ is the random variable defined as the minimum number of $n$ such that $\sum_{i=...
2
votes
0answers
27 views

Non linear regression calculator

Are there any really good non linear regression calculators around the web? Or is something like matlab the best solution? I tried using excel and its solver tool, but it's complete garbage lol. ...
2
votes
0answers
105 views

How to use Taylor series to get $e^x\geq1+x$

I know that from $$e^x=\sum_{i=0}^\infty \left(\frac{x^i}{i!}\right)$$ we can get the inequality $e^x\ge1+x$. But how?
2
votes
0answers
77 views

Evaluating a difficult integral

I need to evaluate this integral $$\int_{0}^{1}\mathrm dx\, x^{m}\exp\left[-k\frac{x}{(1-x^2)^{2/3}}\right],$$ where the real number $k>0$ and $m$ is an integer, $m\geq 0$. Is there any way to get ...
2
votes
0answers
28 views

On the exponential function.

I was rereading the prologue to Real and Complex Analysis by Rudin (in image) and I realized I never really understood why we have the first and second equality after$\sum_{k = 0}^{\infty}\frac{a^k}...
2
votes
0answers
74 views

Level curves of $e^{\alpha z}$

Sketch the families of level curves of $u$ and $v$ for the function $f=u+iv$ given by $f(z)=e^{\alpha z}$, where $\alpha$ is complex. My work so far: \begin{align*} e^{\alpha z} &= e^{(a+ib)(x+...
2
votes
0answers
68 views

Solving $-1=e^a-2e^{av}$ as part of a equation system

Problem Given $f_2(x)=e^{ax-b}+c$ with $x \in \left(0,1\right)$, I am trying to calculate the parameters $a,b,c$ in respect to the following constraints: $$ \begin{align} f_2(0) &= 0 \\ f_2(...
2
votes
0answers
91 views

Tough integral with exp

Can anybody integrate this: $$ \int_0^K e^{i(A\sqrt{\mathstrut k^2+m^2} - Bk)} dk,$$ where $K$, $A$, $B$ and $m$ are real constants? Sorry folks, I didn't realise anybody would be interested in the ...
2
votes
0answers
59 views

Bounds on the product of a matrix exponential and a vector

I have a control system with a state matrix $S = -B^{-1} A \in \mathbb{R}^{n\times n}$, where: $B$ is a strictly positive diagonal matrix $A$ is positive definite $M$-matrix I know that all the ...
2
votes
0answers
52 views

Exponential equations

Let $a,b\geq 1$ be integers and $k=\frac{a}{b}>1$. Solve $$(n+1)^k=n^k+1,\quad n\in\mathbb{Z}.$$ It is clear that $n=0$ is a solution for such equation. I found that if $a,b$ are odd, then $n=-1$ ...
2
votes
0answers
69 views

Residue Theorem on an integral contains a Hankel function and a cosine function

I am trying to solve below integration; $$\int_{0}^{\infty} H_{0}^{1}(pR)\sin(pR)\frac{p}{k^2-p^2} dp$$ here $k,R$ are constants. This is related to the question link. Below shows my approach to get ...
2
votes
0answers
42 views

How to convert this limit involving arctan into an exponential?

As part of a larger proof I'm working on (convergence in distribution of a random variable to a certain cdf), I need to show that: $\lim_{n\to\infty} [(\frac1\pi)(\tan^{-1}(ny)+\frac\pi2)]^n=\exp\{-\...
2
votes
0answers
45 views

Maximum density linear combination chi squares

I have a positive linear combination of chi square variables \begin{equation*} X=\sum_{i=1}^k \lambda_i \chi^2(r_i) \end{equation*} the degrees of freedom satisfy $r_i>1$. I need an upperbound ...
2
votes
0answers
45 views

A relation with limits

Let $t \in \mathbb{R}_+$, $\varepsilon > 0$ and $p_\varepsilon \in [0, 1]$. There is in a book the following relation: $$ \underset{\varepsilon \rightarrow 0}{lim} (1 - p_\varepsilon)^{\lceil t / \...
2
votes
0answers
63 views

A property of exponential of operators 2

Let $X$ be a Banach space. The other day I asked if all bounded operators $A:X\to X$ satisfy the following property: (P): All bounded nonzero trajectories $t\mapsto e^{tA}x$ satisfy $$\inf_{t\in \...
2
votes
0answers
57 views

Why continuous growth (based on e) is being simply scaled to match non-limit cases (limit of the (1+1/n)**n formula)?

The constant $e$ is the maximum exponential growth that is possible when it is done continuously, i.e. when what can be called "continuous breeding" occurs. That is the meaning of the $lim_{n\...
2
votes
0answers
58 views

How to estimate parameters of exponential functions?

I am a new bird to math! I have an expression as follows, $$e^{-ux}+e^{-uy}=z$$ I have at least ten pairs of $(x,y,z)$, so how can I estimate the value of $u$? And how to evaluate my results? Hand ...