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

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2
votes
2answers
45 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
24 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
90 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 ...
2
votes
0answers
34 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
votes
0answers
16 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
236 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
113 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
25 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
112 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 $$\lim_{k\to\infty}\frac{k!}{\sqrt{2\pi k}(k/e)^k}=1.$$ ...
-1
votes
4answers
97 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
60 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
90 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
204 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
51 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
1answer
67 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
97 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
72 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
19 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
98 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
77 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
33 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
151 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 ...
1
vote
3answers
37 views

Solve for x with an expression of independent variable in the exponent

I am helping a friend in research. I wish to solve for x, in this equation. $p, k, m, l, r\text{ and }h$ are all constants. They may vary, depending on the user's input, but they would all be ...
3
votes
4answers
291 views

Obtain real part of complex expression

I must verify if the real part of the following expression $$z = \frac{1 + i}{\sigma \delta \left[ 1 - e^{-(1 + i)t/\delta} \right] }$$ is $$\Re(z) = \frac{1}{\sigma \delta} \frac{1}{1 - ...
4
votes
0answers
183 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 ...
1
vote
1answer
65 views

Describe the Riemann surface:

$$W = \sqrt{1-z^2}$$ I would like hints only. Using @Dr.MV's hint, I get two factors: the first is $$\sqrt{(x-1)+y^2}^{\frac{1}{2}}e^{i\frac{\theta}{2}}$$, which, when we let theta range from 0 to ...
2
votes
0answers
66 views

Derivative of the matrix exponential with respect to its matrix argument

I was trying to find the Frechet derivative of $f = \exp(X)$, where $X \in \mathbb{R}^{n\times n}$ is positive definite. I thought it ought to be $\exp(X)$. I see results where the derivative is with ...
1
vote
3answers
51 views

Show $|\exp(-x/2) - \exp(-y/2)| \leq |x-y|/4$ for $x,y\geq 0$. [closed]

I am trying to show this inequality: $$ \left|e^{-x/2} - e^{-y/2}\right| \leq \frac{|x-y|}{4} $$ for $x,y\geq 0$. I've gotten stuck and could use some kind assistance. Many thanks in advanced!
0
votes
3answers
83 views

Alternative infinite summations that equal $e$

Everyone (and I mean everyone) knows this sum: $$\sum_{n=0}^\infty \frac{1}{k!} =e$$ Are there any lesser known infinite sums that go to e?
1
vote
2answers
83 views

Solving for $x$ : $a^x+b^x=c$

Well the question is to solve for $x$ in $$a^x+b^x=c \tag{a,b,c are constants}$$ Well as of me, I tried to put $\ln{}$ on both sides which does not seem to help. Apart from this I don't seem to ...
0
votes
1answer
29 views

Comparison between exponential and factorial results

I'm developing an algorithm to compare if the result of $n!$ is bigger than $k^m$, but I have problems with big integers, then I need to know if there's some property that I can use to do this without ...
-1
votes
1answer
29 views

The Euler number and exponential function from the property of being own derivative

I watched an MIT video about the Euler number. There they figure it out as follows: The exponential function should be a function that per definition has the property, that it equals to its ...
1
vote
4answers
55 views

Prove that If $0<x<\ln 2$ then $x^2\geq e^x-x-1$

If $0<x<$ln $2$ then $x^2\geq e^x-x-1$ I got this problem while reading a proof. Tried to prove it but failed. $e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$. So $e^x\geq 1+x$ for all $x$ but ...
2
votes
1answer
56 views

Find the Limit: $\lim_{x\to2^{+}}e^{3/2(2-x)}$

$$\lim_{x\to2^{+}}e^{3/2(2-x)}$$ Properties of the Natural Exponential Function: The exponential function $f(x)=e^x$ is an increasing continuous function with domain $\mathbb R$ and range ...
1
vote
3answers
80 views

Prove that $n^a < a^n$ for $a>1$ and $n$ big enough

How can I solve this? I'm trying to prove using limits but it's not working.. Thanks
1
vote
0answers
36 views

What are modulos and how would I be able to use them to solve questions regarding the last digit of a raised power?

When given questions like "What is the last digit of the result to 3^56?", I usually look for a recurring pattern involving smaller powers of 3. In this question for example, the recurring pattern for ...
4
votes
4answers
324 views

Algorithm for rolling an infinite-sided weighted die

If I wanted to have a die that rolled, for example: ...
1
vote
1answer
29 views

Monotonicity of matrix exponential for special matrices

Let $D$ be a matrix having positive off-diagonal values and nonpositive diagonal values such that the row sums are nonpositive and $D$ is invertible. Then $-D$ is an M-matrix. Now decrease some ...
0
votes
2answers
59 views

Abnormal graph curve explanation? Exponential?

I am conducting research on a plot of data. Most appear linear, as expected, but one series was different. The dark red plot (below) is a more accelerated growth curve - and I'm wondering why? Here ...
1
vote
3answers
55 views

$k2^x+2^x=8$, find the possible values of $k$ [closed]

Find all the possible values of $k$ such that equation $$k2^x+2^x=8$$ has a single root. Find the root in the case. Can anyone give some hints for me? I have no idea how to solve it.
0
votes
1answer
23 views

A function for non-linear animation steps (large in the middle, small at the ends)

In a word game for Android I animate movement of letter tiles (for example when user selects "shuffle tiles" or "return tiles from game board" in menu) in a linear way (they have constant velocities) ...
1
vote
1answer
45 views

Why is this true: $1- (1-1/n)^{\varepsilon n} \leq \varepsilon + \mathcal{O}(\varepsilon^2)$

In my lecture notes, the following is written: $$1- (1-1/n)^{\varepsilon n} \leq \varepsilon + \mathcal{O}(\varepsilon^2)$$ as $\varepsilon \rightarrow 0$ and $n$ some fixed constant (non-negative ...
0
votes
1answer
74 views

Closed form for an integral

I am trying to find a closed form for this integral: $\int\limits_{a}^{\infty} \exp(-\frac{b}{x})\exp(-cx)dx$ where a,b,c, are positive constants. Does anyone have any suggestions or can advise? ...
0
votes
1answer
54 views

Solving an exponential function

I have the below exponential function which I wish to solve it for $b$. Other than resorting to the Lambert W function, is there alternative way of representing the solution? $$ \frac{(1+a)(1-b)}{ab ...