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

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3
votes
2answers
231 views

Is there any proof for this formula $\lim_{n \to ∞} \prod_{k=1}^n \left (1+\dfrac {kx}{n^2} \right) =e^{x⁄2}$

Some times ago, In a mathematical problem book I sow that this formula. I don't no whether it is true or not. But now I'm try to prove it. I have no idea how to begin it. Any hint or reference would ...
3
votes
4answers
94 views

Show that the standard integral: $\int_{0}^{\infty} x^4\mathrm{e}^{-\alpha x^2}\mathrm dx =\frac{3}{8}{(\frac{\pi}{\alpha^5})}^\frac{1}{2}$ [duplicate]

In my physics course this standard formula is used a lot without proof so it would be interesting to see a neat proof for it. From a previous thread by me I know the proof for $\int ...
1
vote
1answer
28 views

Exponential Derivative Word Problem

I am having problem with a world problem derivative application question. The number of parasites in the blood after $h$ hours medication is taken is given by the function $p = ...
3
votes
1answer
88 views

When does $\exp\left(\sum_{i=1}^\infty a_i\right)=\prod_{i=1}^\infty \exp(a_i) ?$

Considering the complex logarithm, when do we have $$\exp\left(\sum_{i=1}^\infty a_i\right)=\prod_{i=1}^\infty \exp(a_i) ?$$ I originally wanted to try to prove it by showing $$\lim \prod^N ...
0
votes
0answers
12 views

Partial sum of exponential series strictly increases after certain step (repost)

I apologize for the repost, previous formulation was not precise unfortunately. While trying to show that partial exponential series evaluated at two different ...
0
votes
2answers
57 views

Is it possible to figure out the coefficients of an exponential equation given a certain number of points?

For exponential equations in the form of: $$f(x) = a^x + b^x ,$$ is it possible to solve for a and b if you have a certain number of points? The answers to the similar question here pertain to ...
0
votes
1answer
36 views

Solving special equation [on hold]

How can I find $x$ from the following function while we know that $a,b, c , d$ are constants? $$y= (a b x^{b-1}+ c d x^{d-1}) e^{-ax^{b} - c x^{d}}$$
37
votes
19answers
3k views

Simplest or nicest proof that $1+x \le e^x$

The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or ...
2
votes
5answers
57 views

Show that $\int x\mathrm{e}^{-\alpha x^2}\mathrm dx =\dfrac{-1}{2\alpha} \mathrm e^{-\alpha x^2}$ + Constant

I tried to do this integration by parts and got $\int x\mathrm{e}^{-\alpha x^2}\mathrm dx =\dfrac{-1}{2\alpha} \mathrm e^{-\alpha x^2} +\alpha\int x^3\mathrm{e}^{-\alpha x^2}\mathrm dx$ + constant. ...
1
vote
3answers
182 views

Finding a closed-form formula for a sequence that is defined recursively

$$a_0 = 0, a_1 = 1 \quad \text{ and } \quad a_n = a_{n-1} + 2a_{n-2}\quad \text{ for }n\geq 2$$ a) Find $a_2,a_3,a_4,a_5$ b) Find a closed form-formula for $a_n$ I found the value to be ...
0
votes
1answer
21 views

Expectation of geometric summation of exponentail random variables

We have $\{X_i, i = 1,2,\ldots\}$ as a sequence of independent exponentially distributed rv's with parameter $\lambda$. We also have, $Y =\sum_{i=1}^{N} X_i$. I need to prove that, $Y$ has the ...
2
votes
3answers
35 views

Forming differential equation

I'm trying to get from: $$e^{\lambda t} (\frac{dN}{dt} + \lambda N) = re^{\lambda t} $$ To: $$ \frac {d}{dt}(Ne^{\lambda t}) = re^{\lambda t} $$ However I'm not sure what procedure to use to go ...
0
votes
1answer
24 views

Calculate the matrices $[{A{π \over 4}]}^v$ for all possible values of v

Calculate the matrices $[{A{π \over 4}]}^v$ for all possible values of $v$, when $A(\varphi)=\left(\begin{matrix} \cos\varphi &\sin\varphi\\ -\sin\varphi & \cos\varphi\end{matrix}\right)$ . ...
4
votes
0answers
169 views

Is Every transcendental entire function $f(z) = C + \exp a + \exp b + \exp c $?

Let $z$ be a complex number and let $f(z)$ be any transcendental entire function. Is it true that $f(z) = C + \exp a + \exp b + \exp c $ where $ a,b,c $ are entire functions of $z$ and $C$ is a ...
0
votes
1answer
13 views

Exponential equation with fractions

Solve $6^{5/2}$ $\left(\dfrac{3}2\right)^{-3/2}$ where i get $6^{5/2}$ *$\dfrac{3^{-3/2}}{2^{-3/2}}\cdot$ which i get to $6^{5/2}$ *$\dfrac{2^{3/2}}{3^{3/2}}\cdot$ which leads to ...
2
votes
1answer
215 views

Partial sum of exponential series strictly increases after certain step

While trying to show that partial exponential series evaluated at two different values are strictly increasing provided that sufficient number of terms are applied I stuck at a problem. Given two ...
1
vote
2answers
27 views

Solving an exponential equation that includes division and multiplication

The question is simplify the expression $\left(\dfrac{a^2}{27}\right)^{1/3}\left(\dfrac{64}a\right)^{2/3}$ 1: Multiply on both sides equals $\dfrac{a^{2/3}}{27^{1/3}}\cdot \dfrac{64^{2/3}}{a^{2/3}}$ ...
2
votes
0answers
41 views

Exponential equations

Let $a,b\geq 1$ be integers and $k=\frac{a}{b}>1$. Solve $$(n+1)^k=n^k+1,\quad n\in\mathbb{Z}.$$ It is clear that $n=0$ is a solution for such equation. I found that if $a,b$ are odd, then $n=-1$ ...
-1
votes
2answers
51 views

Solving 4 exponential equations simultaneously [on hold]

These are the 4 equations $$-2=ab^{-1} + c$$ $$-1=ab^0 + c$$ $$1=ab^1 + c $$ $$5=ab^2 + c$$ How would you solve these equations?
4
votes
1answer
102 views

If $\frac{x-1}{e^x-1} = y$ then $x=?$

I have following equation: $$\frac{x-1}{e^x-1} = y$$ I want to solve this equation such that I have the value of $x$ in the term of $y.$ i.e. inverse of the equation
1
vote
1answer
28 views

Determine the order of $\exp Q(z)$ when $Q$ is a polynomial of degree $q$.

I am looking to determine the order of $f(z) = \exp Q(z)$ when $Q$ is a polynomial of degree $q$. I think the order is $q$, but I am struggling to prove it. The definition of order is: An entire ...
6
votes
1answer
124 views

Solution to the functional equation $f(x^y)=f(x)^{f(y)}$

Consider the functional equation problem $$ f: \Bbb{R} \rightarrow \Bbb{R}$$ $$ f(a^b) = f(a)^{f(b)},$$ when $a,b \in \Bbb{R}, a,b \ge 0.$ So far the only solution I have is the trivial $$ f(x) ...
2
votes
2answers
40 views

Complex exponential to real

I'm not yet very good at complex number, so I would appreciate the following insight: How exactly do we arrive from $e^{\pi(1-i)}-e^{-\pi(1-i)}$ to $e^{-π}-e^π$, and why does ...
1
vote
0answers
18 views

Residue Theorem on an integral contains a Hankel function and a cosine function

I am trying to solve below integration; $$\int_{0}^{\infty} H_{0}^{1}(pR)\sin(pR)\frac{p}{k^2-p^2} dp$$ here $k,R$ are constants. This is related to the question link. Below shows my approach to get ...
5
votes
1answer
87 views

Solve $a^x+b^x=c$ for $x$

I need to solve an equation of the form $$a^x+b^x=c$$ with $a,b\in (0,1)$ and $c\in(0,2)$ (and I'm solving for $x\in\mathbb{R}_{>0}$). I know this admits a solution (details below), but it's such ...
10
votes
3answers
1k views

Prove $e^{x+y}=e^{x}e^{y}$ by using Exponential Series

In order to show $e^{x+y}=e^{x}e^{y}$ by using Exponential Series, I got the following: $$e^{x}e^{y}=\Big(\sum_{n=0}^{\infty}{x^n \over n!}\Big)\cdot \Big(\sum_{n=0}^{\infty}{y^n \over ...
1
vote
0answers
32 views

How can we prove $e^{x+y}=e^{x}e^{y}$ by the power series form of exponential function? [duplicate]

How can we prove $e^{x+y}=e^{x}e^{y}$ by the power series $$e^{x}=\sum_{k=0}^{\infty}\dfrac{x^{k}}{k!}\,\,\,?$$ Is there any simple method?
7
votes
3answers
102 views

Integration of exponential functions and cosine function

I am trying to solve the following equation; $$\int_{-1}^{1}e^{i(x+a\cos x)} \, \mathrm{d}(\cos x)$$ or $$\int_{0}^{\pi}e^{i(x+a\cos x)} \sin x \, \mathrm{d}x$$ I tried this in Wolfram Alpha, but it ...
9
votes
1answer
150 views

For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?

Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output. I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and ...
7
votes
2answers
192 views

Show that the series converges, but not absolutely.

Show that the series converges, but not absolutely. $\sum_{n=1}^{\infty}( $exp$(\frac{(-1)^n}{n})-1)$. My Try: Let $a_n=$exp$(\frac{(-1)^n}{n})-1$. I was going to use alternating series test ...
0
votes
0answers
6 views

ARD Kernel - explanation

The following text discusses that the ARD kernel is a regular gaussian kernel but one where $\Sigma$ is diagnonal and one where the $\sigma$'s go to infinity. It seems that the $\kappa$(x,x') would ...
1
vote
2answers
16 views

Effective inter-arrival time converge to mean

I am fairly new to statistics and just recently encountered queueing theory. I have programmed a simulation for a $M/M/1$ queue in which I specify the inter-arrival times and service times. I input ...
0
votes
2answers
43 views

how to solve this complex exponential integration ??

During exercising and example of Fourier Series , I encountered with an integration : $$ \frac{E\omega_o}{4\pi j}\int_{0}^{\frac{\pi}{\omega_o}}\Big[e^{-j\omega_o (n-1)t}-e^{-j\omega_o ...
5
votes
2answers
107 views

Determine all real $x$ for which the series $\sum\limits_{k=1}^\infty\frac{k^k}{k!}x^k$ converges.

Determine all real $x$ for which the following series converges: $$\sum_{k=1}^\infty\frac{k^k}{k!}x^k.$$ You may use the fact that $$\lim_{k\to\infty}\frac{k!}{\sqrt{2\pi k}(k/e)^k}=1.$$ ...
13
votes
4answers
1k views

Function that looks a lot like exponential, but isn't

I'm looking for a continuous function f(x) with the following properties. I've been playing with exponentials, but that doesn't seem to be the answer, although my high school mathematics is a bit ...
4
votes
0answers
58 views

Integral, possibly of Bessel or Exponential form.

I'm working with a hierarchical statistical model, whereby the output of a log-normal distribution affects the argument of an exponential distribution. I need to marginalize, obtaining the following ...
2
votes
4answers
106 views

Evaluate the Integral:$\int\frac{(1+e^x)^2}{e^x}\ dx$

Evaluate the indefinite integral $$\int\frac{(1+e^x)^2}{e^x}\ \mathrm{d}x$$ My attempt: Expand numerator: $$\int\frac{1+2e^x+e^{2x}}{e^x} \, \mathrm{d}x$$ divide $e^x$ by the numerator: ...
-1
votes
4answers
83 views

Direct proof for convexity of $e^x$ [closed]

Is there any direct proof without using second derivative for convexity of $e^x$?
-1
votes
5answers
92 views

Solve exponential equation

I'm dealing with a problem here. I'm trying to solve this exponential equation but I cannot find the solution: $$3^{x-1} + 3^{x-2} + 3^{x-3} + 3^{x-4}\cdot3^{x-5} + 3^{x-6}=364$$ Can anyone please ...
3
votes
1answer
88 views

Integral of Sinc times Exponent of Squared variable

I would like to integrate this in my research: $$\int\limits_{-\infty}^\infty{\frac{e^{i b x^2}\sin{(a x)}}{x}}dx$$ where a and b are both real and greater than zero. If possible, I would like to ...
0
votes
4answers
236 views

How to solve the definite integral $\int_{-4}^{-2}e^{-x}\,dx$?

I'm trying to find the value of the integral $\int_{-4}^{-2}e^{-x}\,dx$ but I just couldn't solve it. Actually I found in a List of integrals that $\int e^x\,dx=e^x+C$ so I concluded: $$ \int ...
1
vote
1answer
43 views

Integral that resembles an exponential integral

$$ I(y;c,\lambda) \equiv\int_{0}^\infty \frac{\lambda c}{x} \exp\left(-\lambda x\right)\exp\left(-\frac{c}{x}y\right)dx$$ where $c,\lambda>0$. Q: Can this integration be made in analytic form ...
-1
votes
3answers
93 views
1
vote
1answer
57 views

Integral exponential and fraction of powers

I am trying to solve the following integral $$ \int_0^y \frac{x^{m-1}}{(1+x)^{m+k}}\, \exp\left(-\frac{m}{\gamma} x \right) \,dx. $$ I tried to look into different books such as Gradshteyn and ...
3
votes
5answers
74 views

Using equation to find value of $1/x - 1/y$

$$\left(\frac{48}{10}\right)^x=\left(\frac{8}{10}\right)^y=1000$$ What is the value of $\frac{1}{x}-\frac{1}{y}$? I have already used that when $48$ divided by $10$ then it becomes $4.8$ and when $8$ ...
0
votes
3answers
50 views

Questions regarding exponential equations

Question:solve $(\sqrt{2}+1)^x +(\sqrt{2}-1)^x=6^{x/2}$ My try:First I was trying to solve it algebrically and tried some things like squaring both sides and tried to simplify but anything didn't ...
2
votes
2answers
58 views

Solving exponential equation

Here is the question:Solve $5^{\frac{x}{2}}-2^x=1$ How i tried:I was just looking at the equation and was trying different values of x and got x=2 .But the way to reach answer was not promising so I ...
2
votes
2answers
97 views

Finding the Sum of a series $\frac{1}{1!} + \frac{1+2}{2!} +\frac{1+2+3}{3!}+…$

I need to find the sum of this series $\dfrac{1}{1!} + \dfrac{1+2}{2!} + \dfrac{1+2+3}{3!}+...$ But somehow I am not even convinced this converges. I tried writing it as $\sum \dfrac{n(n+1)}{2(n!)}$. ...
1
vote
2answers
66 views

Solve this exponential equation: $3^{2x}+\left(\frac{1}{2}\right)^{-x} \cdot 3^{x+1}-2^{2x+1}=0$

I tried solving this equation $$3^{2x}+\left(\frac{1}{2}\right)^{-x} \cdot 3^{x+1}-2^{2x+2}=0$$ by taking the log of both sides, but with no results, what do I do? Sorry if this equation is very easy, ...
1
vote
1answer
17 views

Find a function in the style of $-\tanh(x)$ with a few conditions

I'm searching for a function that looks somewhat like a shifted $-\tanh(x)$-function Through some searching and playing with Wolfram Alpha I managed to shift it in the x-direction, which is partly ...