Articles of exponential function

Proof: $\lim \limits_{n\to\infty}(1+\frac{z}{n})^n = \exp(z)$

I define $\exp: \mathbb C \to \mathbb C$ as $z \mapsto \sum \limits_ {k=0}^{\infty}\frac{z^k}{k!}$. I would like to show that $\lim \limits_{n\to\infty}(1+\frac{z}{n})^n = \exp(z)$. I have a proof for the case $z \in \mathbb R$, but the proof assumes that $\lim \limits_ {n\to\infty}(1+\frac{z}{n})^n$ exists, which is easy to see if $z \in \mathbb R$, but […]

Two exponential terms equation solution

Let $A_i$ and $B_i$ denote constants, I know this equation $$A_1 \exp(B_1x) + A_2x + 1 = 0$$ can be solved using lambert W function. But can I get a general solution of this equation? $$A_1 \exp(B_1x) + A_2 \exp(B_2x) + A_3x + 1 = 0.$$ I searched a lot but did not get an […]

Limit involving tetration

Let the notation be $a^{\wedge\wedge}b = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_{b\,times}$ for tetration. My mentor conjectured the following: Let $n$ be a positive integer, then let $A(n)$ be any function that satisfies $$\lim\limits_{n\to\infty} \left(e^{\frac1e} + \frac1n\right)^{\wedge\wedge}\left[(10 n)^{1/2} + n^{A(n)} + C+o(1)\right] – n = 0$$ where $C $ is a constant. Then $\lim\limits_{n\to\infty} A(n) = \frac1e $ Conjectured by […]

limit of $\left( 1-\frac{1}{n}\right)^{n}$

limit of $$\left( 1-\frac{1}{n}\right)^{n}$$ is said to be $\frac{1}{e}$ but how do we actually prove it? I’m trying to use squeeze theorem $$\frac{1}{e}=\lim\limits_{n\to \infty}\left(1-\frac{1}{n+1}\right)^{n}>\lim\limits_{n\to \infty}\left( 1-\frac{1}{n} \right)^{n} > ??$$

Covariance between squared and exponential of Gaussian random variables

Assuming the random vector $[X \ \ Y]’$ follows a bivariate Gaussian distribution with mean $[\mu_X \ \ \mu_Y]’$, and covariance matrix $ \left[ \begin{array}{cc} \sigma_X ^2 & \sigma_Y \ \sigma_X \ \rho \\ \sigma_Y \ \sigma_X \ \rho & \sigma_Y^2 \end{array} \right]$, I am looking for the expression of $\text{Cov}(X^2, \exp{Y})$. I know there […]

Fit exponential with constant

I have data whic would fit to an exponential function with a constant. So y=aexp(bt) + c Now I can solve an exponential without a constant using least square by taking log of y and making the whole equation linear. Is it possible to use least square to solve it with a constant too ( […]

Is showing $\lim_{z \to \infty} (1+\frac{1}{z})^z$ exists the same as $\lim_{n \to \infty} (1+1/n)^n$ exists

My expanded question: Is showing $\lim_{z \to \infty} (1+\frac{1}{z})^z$ exists as $z$ goes through real values the same as $\lim_{n \to \infty} (1+\frac{1}{n})^n$ exists as $n$ goes through integer values? If not, how much additional work is needed to make the two equivalent? I am asking this because I had posted a question which stated […]

Simply this exponential integral

I am trying to simplify of this: $$\int_{0}^{\infty} \frac{1-e^{-x}}{x}e^{-\lambda x}\,dx.$$ Maybe I should separate these equation into two exponential integral function? But it will ended up with infinite minus infinite? please give me some help or advices, thanks!

Proof of the derivative of $a^x$

This question already has an answer here: Derivative of $a^x$ from first principles 4 answers

Evaluating a double integral involving exponential of trigonometric functions

I am having trouble evaluating the following double integral: $$\int\limits_0^\pi\int\limits_0^{2\pi}\exp\left[a\sin\theta\cos\psi+b\sin\theta\sin\psi+c\cos\theta\right]\sin\theta d\theta\, d\psi$$ I will be satisfied with an answer involving special functions (e.g. modified Bessel function). I tried expanding the product terms using the product-to-sum identities, as well applied other trigonometric identities, to no avail. Can anyone help? I suspect that the solution will involve […]