Questions tagged [exponential-function]

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

529 questions
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Show that a statistical model belongs to exponential family

I have this statistical model: $$f_{j,k}(y)=\frac{\sqrt{j}}{\sqrt{2 \pi}}e^{\sqrt{jk}}y^{-\frac{1}{2}} \text{exp}\left( -\frac{1}{2} (j y + \frac{k}{y}) \right) \quad \quad y>0$$ In this case the ...
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complex numbers. help in proof that $e^{-it}=1/e^{it}$

I need to show that $e^{-it}=\frac{1}{e^{it}}$. but I don't understand what needs to be proven, it seems trivial to me. If anyone could help me. Is the claim true even if t is not real? Thank you
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The mathematics underlying this baroque expression of the double-factorial

On the OEIS page for the double factorial, there are three ways of getting the sequence in PARI. One of them is this curious bit of PARI code: ...
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Solving an expression containing two added exponential functions

I have a problem solving the below equation with respect to $x$: $0.6\cdot \exp(\frac{-40}{x})+0.4 \cdot \exp(\frac{10}{x})=1$ My problem is that I have two exponential functions which are added ...
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How do I prove that the exponential function $e^x$ has gradient $e^x$ from first principles?

Furthermore why is it that $e^x$ is used in exponential modelling? Why aren't other exponential functions used, i.e. $2^x$, etc.? Thank you!
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If $x^x=y^y$ and $x,y>1$, prove that $x=y$

A friend and I have been trying to prove the following: If $x^x=y^y$, where $x,y\ge1$, prove that $x=y$. Intuitively, it's correct, but we haven't been able to prove it. We've tried a few ...
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Equation which should be independent of $x$

An equation has the following form: $$Ae^{-i D x} + Be^{-i E x} = Ce^{-i F x}$$ where $A,B,C \in \mathbb{C}$ and $D,E,F \in \mathbb{R}$ are all constants and $x \in \mathbb{R}$, while $i$ is the ...
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Integration of $x^3 e^x$

I am beginner in calculus and I am struggling with this integral: $$\int x^{3}e^{x}\mathrm{d}x$$ If anyone could give me some hints, any help will be appreciated.
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Limit involving $n$ th root of expression with factorials: $\lim_{n\to\infty}\frac1n\left\{\frac{(2n)!}{n!}\right\}^{\frac{1}{n}}$

I have a problem that finding the values of limit for the following expression. $$\lim_{n\to\infty}\dfrac{1}{n}\left\{\dfrac{(2n)!}{n!}\right\}^{\dfrac{1}{n}}$$ Thank you.
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A limit to infinity: $\lim_{x \to \infty}\ (1+ {\frac{1}{x}})^{x}$ [duplicate]

I've tried substituting ${\frac{1}{x}}$ as $y$ and then I get $\lim_{y \to \ 0}\ (1)^{\frac{1}{y}}$ and that's infinity? $$\lim_{x \to \infty}\ (1+ {\frac{1}{x}})^{x}$$
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Surjectivity of the complex exponential without using $π$

I want to prove that the exponential $\exp\colon ℂ → ℂ^×$ is surjective without using polar coordinates and without even using a definition of $π$. Is there such a conceptional argument? Say, I ...
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Finding out a limit $\lim_{x\rightarrow 0} \frac{\cos x - e^{- \frac{x^2}{2}}}{x^4}$

I have trouble finding $$\large \lim_{x\rightarrow 0} \frac{\cos x - e^{- \frac{x^2}{2}}}{x^4}.$$ I can use L'Hospital's rule, but it seems slightly ineffective there. Can you help me, please?
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Prove the identity between formal power series in t

I am having troubles with proving the above identity. Can I please ask for someone's help? I have spent more than two days on this question and I am kind of exhausted. So I would really appreciate if ...
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Proving that the exponential function is bijective

Prove that $\exp: \mathbb{R} \mapsto (0,\infty)$ is a bijection. Okay, so the first part is really easy: injectivity follows directly from writing the exponential function as a series. Surjectivity.....
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Why does $\lim_{x \to \infty} \big(1 + \frac{1}{x}\big)^x = \lim_{x \to 0} \big(1 + x\big)^{\frac{1}{x}}$?

Is there a way to make sense of that relationship? Could you derive one from the other algebraically? It looks like the first limit approaches $e$ from lower values (for positive $x$ values) whereas ...
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Confusion about Continuous Growth being “a rate that is applied constantly to the amount present at any instant”

(From Morris Kline's Calculus) Talking about continuous compounded interest, he says: "The most important point about the above discussion is that continuous compounding of interest at the rate of ...
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Are the isolated singularities of a holomorphic function $f$ from the same kind as the ones from $e^f$?

Given any meromorphic function $f: U \to \mathbb{C}, U \subseteq \mathbb{C}$ open, I want to prove or disprove the statement: for any isolated singularity $a \in U$, the function $g(z) = e^{f(z)}$ has ...