# Tagged Questions

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

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### $\int_{- \infty}^{\infty} \frac{f(x)}{1+\exp{g(x)}}dx=\int_{0}^{\infty} f(x) dx$ 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 ...
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### 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 ...
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### 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?
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### 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.
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### 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 ...
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### 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 ...
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### 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'''...
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### 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 ...
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### 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 ...
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### 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 $n=-1$ ...
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### 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 ...
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: $\lim_{n\to\infty} [(\frac1\pi)(\tan^{-1}(ny)+\frac\pi2)]^n=\exp\{-\... 0answers 45 views ### Maximum density linear combination chi squares I have a positive linear combination of chi square variables \begin{equation*} X=\sum_{i=1}^k \lambda_i \chi^2(r_i) \end{equation*} the degrees of freedom satisfy$r_i>1$. I need an upperbound ... 0answers 45 views ### A relation with limits Let$t \in \mathbb{R}_+$,$\varepsilon > 0$and$p_\varepsilon \in [0, 1]$. There is in a book the following relation: $$\underset{\varepsilon \rightarrow 0}{lim} (1 - p_\varepsilon)^{\lceil t / \... 0answers 63 views ### A property of exponential of operators 2 Let X be a Banach space. The other day I asked if all bounded operators A:X\to X satisfy the following property: (P): All bounded nonzero trajectories t\mapsto e^{tA}x satisfy$$\inf_{t\in \... 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\...
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 ...