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

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-1
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0answers
24 views

derivative of exponential function proof- [on hold]

I will post link because stackE won't accept size or type of file. This is part of proof for derivative of exponentinal function. 1.part http://postimg.org/image/iz97da581/ How do they get bottom ...
1
vote
1answer
34 views

Finding the Zeros

A word problem gives this cost equation and asks to find the x where the average cost is minimized, to do so, I need to solve for average cost and derive it and then set it to zero and solve; $c(x) = ...
1
vote
2answers
63 views

Why does $\lim_{x\to0^{-}} \mathrm {Im}\left( \mathrm \ln \left(x\right)e^x\right)=\pi$?

Why does $$\lim_{x\to 0^{-}} \mathrm {Im} \left( \ln\left(x\right) e^x\right)=\pi$$ Obviously this is no coincidence. I was thinking maybe this has to do with Euler's formula, but I don't see how the ...
0
votes
1answer
44 views

How to integrate $\int_{l1}^{l2}\frac{e^{\pm i a x}}{\sqrt{bx^2+cx+d}}dx$

I have the above mentioned integral $$ \int_{l_1}^{l_2}\frac{e^{\pm i a x}}{\sqrt{bx^2+cx+d}}dx $$ which I want to solve. I expect some special functions in its solution, but so far I am out of ...
2
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1answer
32 views

Evaluating a limit I

Consider the limit \begin{align} \lim_{x \to \infty} \left[ \frac{(x+a)^{x+1}}{(x+b)^{x}} - \frac{(x+a-n)^{x+1-n}}{(x+b-n)^{x-n}} \right]. \end{align} It is speculated that the resulting value is ...
4
votes
0answers
33 views

Generalized Trigonometric Functions in terms of exponentials and roots of unity

I am trying to come up with generalized trigonometric functions using the exponential definition that we use today for the trig functions sine and cosine $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}; \cos x ...
3
votes
1answer
42 views

Singularity type of $\frac{1}{z} e^{-\frac{1}{z^2}} $

I've been asked to compute the singularity type of $f(z) := \frac{1}{z}e^{-\frac{1}{z^2}} $. Here's my reasoning: $$ \frac{e^{-\frac{1}{z^2}}}{z} = z^{-1} \sum_{n=0}^\infty \big( -z^{-2} \big)^n ...
-2
votes
0answers
21 views

Scrabble/words with friends [on hold]

How many letter combinations are possible with 7 tiles? Just the math answer please, 7 tiles in 7 slots, how many different combinations? Thank you :)
1
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1answer
33 views

“Time until arrival/departure” in a Poisson process…

Customers are served at a bank with the following process. While there is at most one customer in the bank, there will be only one person teller at a window. If a second customer comes into the ...
1
vote
1answer
39 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
89 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
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0answers
19 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 ...
3
votes
4answers
99 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 ...
0
votes
1answer
38 views

Solving special equation [closed]

How can I find $x$ from the following function while we know that $a,b, c , d$ are constants? $$y= (a b x^{b-1}+ c d x^{d-1}) e^{-ax^{b} - c x^{d}}$$
2
votes
5answers
62 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. ...
2
votes
3answers
37 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
4answers
80 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
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 ...
1
vote
2answers
28 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 $\dfrac{a^{2/3}}{27^{1/3}}\cdot \dfrac{64^{2/3}}{a^{2/3}}$ ...
0
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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 $A(\varphi)=\left(\begin{matrix} \cos\varphi &\sin\varphi\\ -\sin\varphi & \cos\varphi\end{matrix}\right)$ . ...
2
votes
0answers
44 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 $n=-1$ ...
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votes
2answers
51 views

Solving 4 exponential equations simultaneously [closed]

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?
2
votes
2answers
43 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
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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
88 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 ...
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?
0
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0answers
8 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 ...
2
votes
1answer
229 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 ...
7
votes
3answers
105 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 ...
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 $$\lim_{k\to\infty}\frac{k!}{\sqrt{2\pi k}(k/e)^k}=1.$$ ...
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votes
4answers
85 views

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

Is there any direct proof without using second derivative for convexity of $e^x$?
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 ...
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 ...
1
vote
1answer
44 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 ...
7
votes
2answers
198 views

Show that the series $\sum\left(\exp\left(\frac{(-1)^n}{n}\right)-1\right)$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 ...
-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 ...
13
votes
4answers
2k 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 ...
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
59 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 ...
-1
votes
3answers
95 views

How to prove this inequality $(\frac{n+1}{e})^{n} < n! < e(\frac{n+1}{e})^{n+1}$? [closed]

$\Bigl(\frac{n+1}{e}\Bigr)^{n} < n! < e\Bigl(\cfrac{n+1}{e}\Bigr)^{n+1}$
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 ...
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 $\sum \dfrac{n(n+1)}{2(n!)}$. ...
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 $8$ ...
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: ...
2
votes
1answer
32 views

How do I find a point on a graph which is equal on both the axis?

I have the equation $ 10^{x-0.7711} = x $. In order to find x, I thought that I'll graph the equation, and the point where x = y, will be the answer. How do I do this? Or is there any other way to ...
1
vote
1answer
38 views

How to prove $(I-e^{At})^{-1}$ only contains positive element?

Given a symmetric $N\times N$ matrix $A$, with eigenvalues $-x_1,-x_2,-x_3,\dots,-x_N$ and $x_1,x_2,\dots,x_N >0$. $A$ is known as a negative-definite matrix. We can diagonalize $A$ as $A = ...
8
votes
6answers
146 views

To show that $e^x > 1+x$ for any $x\ne 0$ [duplicate]

$$e^x>1+x$$ is what I want to show. So let's define a function: $$h\left(x\right)=e^x-x-1$$ and investigate its derivative: $$h'\left(x\right)=e^x-1$$. Easy to see that at $x=0$ it has a ...