Questions tagged [exponential-function]

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

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Solve the differential equation that define exp(x)

In the wikipedia page for the exponential function in the "formal definition" section I found this statement: Solving the ordinary differential equation $y'(x)=y(x)$ with the [initial ...
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e{ } and e( ) Notation

I was reading An Introduction to the Theory of Numbers by Hardy and Wright, and on pages 66 and 67, I encountered the notation, as following What does the $e( )$ and $e\{\}$ notation mean? Any help ...
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Is the intersection of a function and its inverse on the line $y=x$ equivalent to the function being strictly increasing? [closed]

Is the following statement true? The functions $f$ and its inverse $f^{-1}$ intersect each other on the line $y=x$ if and only if $f$ is a strictly increasing function. I think this statement is not ...
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Closed form of $f(n) = \prod_{m=2}^{n-1}( e^{\pi i n/m} - e^{-\pi i n/m})$

For context, I am not a mathematician, but I like to do explore some math concepts a couple of times a year. I have been playing with an idea for the past few years or so and a while back I asked this ...
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Question on exponential functions and drawing suitable lines on the graph

This question is from an exercise in an Edexcel Further Pure Mathematics book. Parts a, b, and c were fairly easy for me. For part a, I found the corresponding values of y using a calculator and in ...
1 vote
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Examples of expansions of the exponential of a sum of two matrices [closed]

The exponential function of a matrix is fundamental in mathematics, physics and beyond. One can define it using the power series $$e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$ For any matrix $M$ ...
1 vote
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What to consider when taking kth root on both sides of equality

Say I have the following expression: $10^{l} = a^{k}$ If I take the kth root of both sides, does that mean we get: $10^{\frac{l}{k}} = a$ We don't have to consider anything with plus or minus?
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How to calculate $\int_{0}^{\infty} x^n(1+x)^n e^{-ncx^2}\text{d}x$

As the title mentioned, I want to calculate $$\int_{0}^{\infty} x^n(1+x)^n e^{-ncx^2}\text{d}x,$$ where $n$ is a positive integer, $c$ is a positive real number in the ...
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Can a parabola be exactly replicated using exponentials?

I came across this interesting artifact of a problem I was helping a friend solve where I used Euler's exponential forms of sine and cosine which later became of the form $e^{ix} + e^{-ix}$ on one ...
1 vote
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Understanding the domain of a function raised to another function

So the question is to find the sum of all real solutions to the equation $\left(x^{2}-5x+5\right)^{x^{2}+4x-60} = 1$. The solution intends us to take 3 cases: The first case is to set the base equal ...
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1 vote
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Problems with understanding Dirac Delta Properties [closed]

I need to check equation ${\bf 6.15}$ in an $\tt arxiv$ paper for my work, but I don't get why the Dirac Distributions change and why there is that exponential ...
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What formula would I use to calculate total income over $x$ years with a fixed salary increase rate.

I'm wanting to calculate how much total money I would have earned over x amount of years, but while accounting for increase in pay at the end of each year (or maybe even continuous?) Lets say for ...
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Convergence rate of $(1+x/n+\dots)^n$

It is a well known fact that $$\lim_n (1+x/n+O(n^{-3/2}))^n=e^x$$ For example, this is a key step in the standard proof of the central limit theorem. What can we say about the rate of convergence of ...
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Given $\epsilon > 0$, can we find a smooth function $f:[0,\epsilon]\rightarrow [0,1]$ satisfying the following conditions? $f(0)=1$ and $f(\epsilon) =0$. $f^{(n)}(0+) = f^{(n)}(\epsilon-)=0$, i.e.,...