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# Questions tagged [exponential-function]

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

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138 views

### When is the exponential map between matrices injective?

Consider a complex matrix $A$. When does $e^A=e^B$ imply $A=B$? Is there any general statement that can be made as to when this holds? It is clearly not true in general, a trivial example being when ...
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### Expectation normal distribution

I am trying to solve the following exercise: $$\mathbb E[e^Y\textbf 1_{X<T}]$$ where, X and Y $\sim \mathcal N(0,\sigma)$, $\sigma = 0.5$ $\operatorname{cor}(X,Y) = \rho =-0.98$ $T=-3$ I solved ...
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### Understanding solution to $y' = y$ and exponential distribution

My Understanding: I would derive the exponential random variable as follows: I consider an experiment which consists of a continuum of trials on an interval $[0,t)$. The result of the experiment ...
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### Simple form for improper integral $\int_0^{\infty}{x^{1/x-x}dx}$

I've been struggling a while now trying to find an alternate form for $$G=\int_0^{\infty}{x^{1/x-x}dx}$$ in function of known constants, (like $e$,$\pi$,etc.) Having looked at the problem, I don't ...
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### Decisions on the order of integration with double integrals (when Deriving PDF via CDF) (Bank Problem)

Consider the following problem: Gandalf, Saruman and Radagast go to a bank together. There are two open counters which Gandalf and Saruman immediately go to get their service. Radagast goes to the ...
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### How to invert a simple exponential growth formula

I think this is simple but my math skills are limited. I have a basic exponential growth formula: $$y=x \cdot (1-p)^n$$ and I have $y$ and $x$ and $n$ values and I need value of $p$. Then when I solve ...
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### Proving that any continuous homomorphism of $\mathbb{R}/(2\pi\mathbb{Z})$ int0 $T$* is neccesarily an exponential function

This is an exercise form Katznelson's book on Harmonic Analysis, so I want to solve it using his hint. T* here denotes the multiplicative group of units of complex numbers of unit norm. That is to ...
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### 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 ...
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### Hidden Markov Model, transition probabilities which are modeled with an exponential distribution

I'm looking at implementing an algorithm described in a paper, but I'm having trouble understanding how the transition probabilities for a Hidden Markov Model are defined. In the first sections, I ...
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### 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 $b$ ...
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### Is the function $f(x)=1^x=1$ considered an exponential function?

I am confused about the following: The exponential function (by definition) is a function of the form $f(x)=a^x$ where $a>0$. However, when $a=1$, we get the constant function $f(x)=1^x=1$. Is the ...
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### Lie group - exponential Diffeomorphism

Let $G$ be a nilpotent, connected simply connected Lie group and $\mathfrak{g}$ its Lie algebra. It is known that the exponential map $\exp$ is a diffeomorphism. Now let $\mathfrak{g}_0$ be a Lie ...
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### Sigmoid Function Question

Ive been trying for well over a week to try to understand how to use a simple sigmoid or logistic function works. Specifically I'm trying to understand how to build proper polynomia parameters for ...
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### A question on Exponential Equation

I came across the following question a few week ago (Exponential equation+derivative): Solve $3^x+28^x=8^x+27^x$. The answer for the above question is 0 and 2. I generalized the question, as ...
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### 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-...
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### The Lie exponential map and commuting elements

By Baker–Campbell–Hausdorff formula, if $[v,w]=0$ for $v,w$ in the Lie algebra $\mathfrak g$ of a Lie group $G$ then $\exp(v)$ and $\exp(w)$ commute in $G$. Does anyone know a reference or a method ...
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### 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}{2}...