Questions tagged [exponential-function]

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

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183 views

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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4answers
47 views

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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1answer
229 views

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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5answers
109 views

Sum of $\cos\theta + \frac{1}{2}\cos2\theta + \frac{1}{4}\cos3\theta + \dots$

In a question I am given the sequence $e^{i\theta}+\frac{1}{2}e^{2i\theta}+\frac{1}{4}e^{3i\theta}+\dots$ I can show that the sum to infinity is $S_\infty=\frac{2e^{i\theta}}{2-e^{i\theta}}=\frac{2\...
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1answer
74 views

Expectation with a variable that satisfies uniform distribution

Do we have step by step proof for this expression which is concerned with the expectation of an exponential function with a variable (R) that satisfies uniform distribution. I have been trying to ...
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2answers
64 views

How to prove that $\lim\limits_{\epsilon\rightarrow 0}\int_{-\infty}^{\infty}f_\epsilon(x)g(x)dx=g(0)$ (Dirac delta function))

I'm currently studying the Dirac delta function using a textbook which unfortunately provides only partial solutions to its explanations. Why does $\lim\limits_{\epsilon\rightarrow 0}\int_{-\infty}^{\...
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1answer
42 views

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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4answers
3k views

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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3answers
74 views

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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1answer
46 views

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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2answers
135 views

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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5answers
389 views

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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2answers
228 views

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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1answer
76 views

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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5answers
55 views

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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2answers
137 views

Find the sum of the roots of the exponential equation

The equation $$2^{333x - 2} + 2^{111x + 2} = 2^{222x + 1} + 1$$ has three real roots. Find their sum. I'll simplify it first as: $$\frac{1}{4}2^{333x} + (4)2^{111x} = (2)2^{222x } + 1$$ Let $u = ...
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1answer
81 views

How to solve the following exponential equation?

How to solve the following exponential equation? $h_1 = x - yq_1^z $ $h_2 = x - yq_2^z$ $h_3 = x - yq_3^z$ here $x$, $y$, $z$ are unknown and $h_1$, $h_2$, $h_3$, $q_1$, $q_2$, $q_3$ are constants....
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1answer
634 views

Length of a very basic exponential curve

I have the beginning points (0,1) and end points (180, 141.732) of a curve. The function I am currently using is f(x) = Ae^kx. However, when deriving the original function, I end up with 0 (from ln(1)...
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1answer
64 views

How can I show the following trigonometric function [closed]

How can I show that $$|\frac{18k(a^{4}-z^{4})}{82a^{2}z^{3}-9z(a^{4}+z^{4})}|=\frac{18k\sin(2\theta)}{a(41-9\cos(2\theta))}$$ Where $z=ae^{i\theta}$, this the point I've gotten to in a calculation ...
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4answers
91 views

$f:\mathbb R \to (0,\infty)$ defined by $f(x)=e^x$. Describe its inverse.

How do I go about describing it? Well first is the inverse $e^{-x}$ or $\ln(x)$? Additionally, since I have no clue how to solve these problems as I am probably overthinking them... $f:\mathbb R\to ...
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1answer
115 views

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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1answer
2k views

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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2answers
116 views

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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0answers
47 views

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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1answer
76 views

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 ...
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1answer
121 views

Definition of Derivative And Exponential Functions

Given $f(x) = 5^{3x}$. Find $f'(x)$ using definition of a derivative. The definition of the derivative of $f(x)$ is $f'(x) = \lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$ The derivative of $f(x) = 5^{...
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1answer
28 views

Integrate the solution of a the Matrix differential equation

I have: $\dot{\textbf{x}}=A{\textbf{x}}$ where A is a nxn matrix. This equation has solution: $\textbf{x}(t)=e^{\textbf{A}t}\textbf{x}(0)$ A book i'm reading states that: $\textbf{x}(t_{2})=e^{\...
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1answer
98 views

Link between $\lim \limits_{n \to \infty} (1+{1/n})^n$ and $\lim \limits_{n \to \infty} (1+{x/n})^n$

I understand that the intuition behind $e = \lim \limits_{n \to \infty} (1+{1/n})^n$, which can be understood as a continuous interest of 100% over time. However I'm having troubles understanding $e^x ...
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1answer
72 views

Solve for $x$ (both 2 values as plotted on graph):

find a way to solve for x: $$ 2^x = x + 5$$ You can easily see one of the values is $3$. If you plot this in a graph, with $y=2^x$ and $y=x+5$, you'll see it has 2 values. If, in a calculator ...