# Questions tagged [exponential-function]

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

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### Evaluating tricky definite complex integral and getting numerical result as a single complex number

For given constants $a\in \mathbb{R}$ and $b \in \mathbb{R}$ and I would like to numerically evaluate the following complex integral $$\int_{-1}^{1} (1-|x|) \exp(ia(b+x)^2) \, d\mathrm{x}.$$ I am not ...
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### How to prove exponential functional identity knowing that it is a solution to a first order ODE and knowing its Taylor expansion

Establish the identity $$E(ax)E(bx) = E[(a+b)x]$$ knowing that $y = E(px)$ satisfies $y' - py = 0$ and $E(px) = \sum_{n=0}^\infty\frac{(px)^n}{n!}$ An additional hint the textbook gives ; "...
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### Finding smallest number returned by an integral [closed]

I wanna preface this: I never formally learned integrals and differentials So, I'm currently helping a friend study for an exam, however... there's a specific exercise here, which is kinda out of my ...
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### Can I rewrite an exponential with product of powers as product of two terms?

I have what I thought was a relatively simple problem but cannot quite pinpoint the answer. I want to do a regression of something like $a \times \mathrm{e}^{-b/x}$. However, I can only regress the ...
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### Which holomorphic functions have constant argument on rays from the origin? On circles centered at the origin?

Multiples $$f(z) = c z, \qquad c \in \Bbb C \setminus \{0\},$$ of the identity function on $\Bbb C \setminus \{0\}$ trivially all satisfy the following special condition: Condition A: All points on a ...
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### Exponential function using binomial approach and approximation by power series, both using python. Why different results?

Why using power serie the result was so different ? I imagine it happened due to the truncation, I used 20 terms, are more terms needed ? EDITED: The code is below ...
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