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

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2
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1answer
37 views

Computing the Taylor expansion of the square root of cos(z),

Let $\large f(z)=\sqrt{cosz}$ with the branch of the square root chosen so that $f(0)=1$. Consider the power series expansion of $f(z)$ in powers of $z$. Part 1) Compute the first three non-zero ...
2
votes
2answers
73 views

Complex Analysis Inequality Involving Exponential

I am trying to prove the following result: For $0<|z|<1$, $\frac{|z|}{4}< |e^z - 1| < \frac{7|z|}{4}$. My attempt : First, I converted the problem to (after mult. by 4): $$|z|< ...
1
vote
3answers
90 views

$\lim\limits_{n\to\infty}$ $(1+ {2\over n})^n$ = $e^2$

If I set N=$2\over n$, the equation becomes $(1+ N)^{2\over N}$, which I can take the natural log of and work down until I reach $e^2$. However, my teacher wants me to use subsequences, starting with ...
1
vote
1answer
32 views

Polynomial with variable in the exponent.

How do I solve a polynomial with the variable in the exponent? For the following question how would I find n? I tried to log both sides but that failed. Note: Both exponents should be negative.
0
votes
1answer
34 views

Are there shortcuts for predicting whether an exponential function will evaluate to a positive or negative value?

I'm a software architect, working on some algorithms involving unique storage formats for large numbers. The function I'm working with is:$$y = \frac{mx^p+b}{d}$$ ... where $p$ and $d$ would ...
0
votes
1answer
23 views

Does $\overline {\lim\sum_{n=0}^{k}\frac {z^n}{n!}}=\lim\overline {\sum _{n=0}^{k}\frac{z^n}{n!}}$?

A book I'm reading claims that $\overline{Exp(z)}= Exp(\overline z)$ where the exponetial function is defined as $\sum_ {n=0 }^{\infty } \frac{z^n } {n! } $ Verifying this would mean verifying that ...
0
votes
1answer
31 views

Sum of probability density functions, exponential distribution

Let $ X_1 $ and $ X_2 $ be iid with pdf $$ f(x)=\lambda e^{-\lambda x}, \quad \lambda>0, \quad x \in (0, \infty)$$ Find the density function of $Z = X_1+X_2$. I tried to solve it using this ...
2
votes
2answers
38 views

Definition for the complex exponential function

We define the exponential function as $\exp(z)=\sum\limits_{j=0}^\infty= \frac{z^j}{j!}$ for all $z\in \mathbb{C}$. Lets now compute $\exp(0)$, then we would have to calculate $0^0$ which undefined. ...
0
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0answers
19 views

Simulate exponential growth on user database

I am currently working on a project where I would like to plot a graph of the cumulative sum of user registrations per day. To test this, however, I need to simulate userdata. I am looking for a ...
0
votes
1answer
40 views

Efficient extraction of coefficient from generating function

In Mathematica I give input for generating function: Series[Exp[x + x^2/2], {x, 0, 6}] It gives output: $$1 + x + x^2 + \frac{2x^3}{3} + \frac{5x^4}{12} + ...
2
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0answers
16 views

Non linear regression calculator

Are there any really good non linear regression calculators around the web? Or is something like matlab the best solution? I tried using excel and its solver tool, but it's complete garbage lol. ...
0
votes
2answers
45 views

Does the definition of $\exp$ as the solution to $y'=y, y(0)=1$ allow us to actually calculate its value at a given point?

Let $\exp:\Bbb R\to \Bbb R$ denote the solution of the ODE $$ y'=y, \quad y(0)=1. $$ Say I want to calculate $\exp(x_0)$ for some given $x_0$, is there any way to do this without using the other ...
3
votes
3answers
102 views

Prove that $\lim \limits _{n \to \infty} \left( n - \frac 1 {e^{\frac 1 n} - 1} \right) = \frac 1 2$

Limit of reciprocal of nth root of e minus 1 is linear I somehow got around to needing to know the value of $\frac 1 {e^{\frac 1 n} -1}$, and noticed that it seems very linear: And that the ...
1
vote
1answer
79 views

How can I prove without using a calculator that $\frac{1}{e} > \frac{\ln \pi}{\pi}$? [duplicate]

Without using a calculator. I can see that $\ln \pi$ is close to $1$ but a little bit greater... Since $e$ is less than $\pi$, $\frac{1}{e}$ has to be a larger number. I don't understand how someone ...
0
votes
1answer
17 views

The radioactive polonium decays to half of its original amoun every 159 days

The radioactive polonium decays to half of its original amount every 139 days (i.e. its half-life is 139 days). If your sample will not be useful to you after 78% of the radioactive nuclei present on ...
1
vote
1answer
15 views

Specific form of an exponential function given two points and a slope

Given a general equation for an exponential function: $$y=Aa^{-x}$$ I would like to find its specific form that would conform the conditions below: it starts from hight $y(0)=H$ at distance $w$, ...
1
vote
2answers
31 views

Inverse of an exponential function

I am having difficulties forming the inverse of this $f(x) = 3 \cdot2^{3x+1} \cdot 5^{3x-1}$. What I have done so far: $3 \cdot 2^{3y} \cdot 2^1 \cdot 5^{3y}\cdot5^{-1} \Leftrightarrow 3\cdot 2\cdot ...
2
votes
4answers
37 views

Equation with different bases (exponential)

I seem to be stuck with the following equation right here: $$2^x + 2^{x+1} = 3^{x+2} + 3^{x+3}$$
0
votes
3answers
81 views

A proof of $=e^{A+B}=e^{A}e^{B}$ using ODEs

I want to prove that $=e^{A+B}=e^{A}e^{B}$ for commuting matrices $A,B$ using differential equations. I found a proof here: LINK Here is how the proof goes: Given a square matrix $M$, the ...
6
votes
1answer
146 views

On $e^{5x}+e^{4x}+e^{3x}+e^{2x}+e^{x}+1$

Define the following, $$F_2(x) := \frac{1}{2}+\frac{(2x)}{1!} B_2\Big(\tfrac{1}{2}\Big)+\frac{(2x)^2}{2!}B_3\Big(\tfrac{1}{2}\Big)+\frac{(2x)^3}{3!}B_4\Big(\tfrac{1}{2}\Big)+\dots $$ ...
1
vote
2answers
29 views

Simplying $ie^{it}(1+e^{-it})^n$

Need some help simplifying $$ie^{it}(1+e^{-it})^n$$ where n is an integer, so I can integrate it between $0$ and $2\pi$ I tried using De Moivres Theorem but the 1+ didn't allow me too
4
votes
3answers
104 views

Do the polynomials $(1+z/n)^n$ converge compactly to $e^z$ on $\mathbb{C}$?

The question is Do the polynomials $p_n(x)=(1+z/n)^n$ converge compactly (or uniformly on compact subsets) to $e^z$ on $\mathbb{C}$? I thought about expanding $$p_n(z)=\sum_{k=0}^n ...
0
votes
0answers
9 views

Simplifying $e^{-i x\log n}+e^{i x\log n}(2^{{1 \over 2}+ i x}\pi^{-{1 \over 2}+i x}\cos({\pi\over 4}-{{\pi i x}\over 2})\Gamma({1\over 2}-i x))$?

I'm working with an equation that has the following term in it: $$e^{-i x\log n}+e^{i x\log n}(2^{{1 \over 2}+ i x}\pi^{-{1 \over 2}+i x}\cos({\pi\over 4}-{{\pi i x}\over 2})\Gamma({1\over 2}-i x))$$ ...
2
votes
2answers
41 views

Minimising Function, derivative with exponentials

First of all, I apologise for not giving a more descriptive title. I really do not know how to word it. I'll go straight into the meat of the question. If a function $$h(x)=\frac{e^x-1}{x^5}$$ is to ...
2
votes
1answer
35 views

Derivatives question on partial derivatives

If $z=f(x,y)$ and $x=e^u \cos v$, $y=e^u \sin v$ then show that $y \frac{dz}{du} + x \frac{dz}{dv} = e^{2u} \frac{dz}{dy}$
0
votes
2answers
62 views

Let $\mathbf A$ be a matrix such that $\mathbf A^2=-\mathbf I$. Prove that $\exp(\varphi\mathbf A)=\mathbf I\cos{\varphi}+\mathbf A\sin{\varphi}$

Let $\mathbf A$ be a matrix such that $\mathbf A^2=-\mathbf I$. Prove that $\exp(\varphi\mathbf A)=\mathbf I\cos{\varphi}+\mathbf A\sin{\varphi}$ This is my attempt: $$\mathbf A^2=-\mathbf I ...
0
votes
1answer
26 views

Derivative of exponential functions

I'm having trouble for the derivative of this exponential function which looks difficult. I have used the quotient rule and chain rule to solve for $0.6^x$ but its still wrong. $$P(x)= ...
1
vote
1answer
20 views

Closed-form solution to transcendental equation with exponential function?

I am dealing with an optimization problem in which I want to find $x$ that minimizes $y(x)$. Instead of resorting to numerical optimization, I would like to find a closed-form solution. So, after ...
1
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0answers
40 views

Integrate exponential with modulus

I need to prove the following integral relation: $$\lim_{T \to (1-i \epsilon) \infty} \mathrm{\int_{-T}^T dt_1 dt_2 e^{-ia |t_1|} \cdot e^{-ib |t_2|} \cdot e^{-ic |t_1-t_2|}} = \frac{-2}{(a+b)(b+c)} + ...
1
vote
2answers
41 views

Prove the following $e^{iAx} = \cos (x)I + i \sin (x) A$

the question says as following, Let x be a real number and A a matrix such that $A^2 = I$. Show that the $e^{iAx} = \cos (x)I + i \sin (x) A$. my problem is that I don't know how to deal with ...
1
vote
1answer
29 views

$e^{-\frac{1}{x}}e^{-\frac{1}{1-x}}$ in 3D

I have the function $f(x) = e^{-\frac{1}{x}}e^{-\frac{1}{1-x}}$, which produces this graphic: What should $f(x,y)$ be to look like a 'hill', i.e. $f(x)$ spinned about vertical axis?
0
votes
1answer
31 views

Is it possible to interpolate $e^n$ in more than one way?

The most basic definition of exponentiation is repeated multiplication, $$e^n = e \cdot e \cdot \cdot \cdot \cdot e$$ $n$ times However, if $n$ is a rational number such as $2.4$, this ...
2
votes
4answers
60 views

Prove that $\lim_{x \to \infty}\big(\frac{x}{x-1}\big)^x$ is also $e$.

Trying to make sense out of the idea that $100\%$ continuous decay is $\frac{1}{e}$, I thought about this: You can express $1+\frac{1}{x}$ as $\frac{x+1}{x}$, such that $\big(1+\frac{1}{x}\big)^x = ...
1
vote
2answers
25 views

Intuition: Why is continuous decay expressed as the inverse of the equivalent continuous growth rate?

I understand $e$ as $\lim_{n \to \infty} \big(1+\frac{1}{n}\big)^n$. I also (finally) understand the idea that continuous growth is "a rate that is applied constantly to the amount present at any ...
0
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0answers
14 views

Are the interest accrued formula the same

Relating to my other post on Stack Exchange http://money.stackexchange.com/questions/56477/correct-way-to-calculating-interest-accrued-with-leap-year I'm wondering is example 1 and example 3 ...
0
votes
0answers
8 views

sample size calculation for count data

I have a plan to see some treatment effect in several projects in my company where I will compare the average number of errors now and after the treatment. So what I know from current situation is ...
-1
votes
2answers
54 views

Euler identity: why isn't “e” a “number”?

$\pi$ is a real number $\mathbb R$ and can be calculated using an infinite product. As far as I know, $\mathrm{e}$ is a real number $\mathbb R$, too. There is an exponential function which is ...
1
vote
1answer
25 views

$|\exp(2\pi i mh)-1|<2^{1-\gamma}(2\pi)^{\gamma}|m|^{\gamma}h^{\gamma}$?

im reading a book about fourieranalysis and found this inequality ... $|e^{2\pi imh}-1|\leq \min(2,2\pi|m|h)\leq 2^{1-\gamma}(2\pi)^{\gamma}|m|^{\gamma}h^{\gamma}$ with $m\in\mathbb{Z}$ and ...
3
votes
1answer
32 views

domain of $x^x$

What will be the domain of $f(x)=x^x$? I have asked this to some teachers, they say that the domain is set of all nonnegative real numbers. It is true that there are infinite negative numbers for ...
0
votes
0answers
24 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 ...
0
votes
1answer
35 views

Solve slope intercept equation for two points and the maximum starting value?

I have two points (x2,y2) and (x3,y3) that represent points in an exponential decay curve of discounted cash flows (x2 is less than x3): My question is: What is the decay curve equation for the ...
1
vote
1answer
21 views

Taking the logarithm of $e^{-x}<b$

If I have an inequality as $e^{-x}<b$ where $b,x$ are positive , can I take the logarithm on both sides and say, $-x<=ln(b)$
1
vote
2answers
49 views

Solve exponenital integral equation

$$\frac{1}{\sqrt{2\pi}\sigma_1 }\int_x^\infty\exp(-\frac {t_1^2-1}{2\sigma_1^2})dt_1 + \frac{1}{\sqrt{2\pi}\sigma_2 }\int_x^\infty\exp(-\frac {t_2^2-1}{2\sigma_2^2})dt_2 = a $$ $$\sigma_1 , \sigma_2 ...
7
votes
4answers
91 views

A function with a property $f(x+y)=f(x)f(y)$

A function with the property $f(x+y)=f(x)f(y)$ is well known exponential function, $f(x)=a^x$. My question is, how do you prove if there is no other function with this kind of property? Edit: I ...
3
votes
3answers
61 views

Random Walk And Stochastic-Processes

Assume that $P(X_i = 1) =1/2, P(X_i =-1)= 1/4,\text{ and }P(X_i = 0)=1/4$. Consider the random walk starting at 1 given by $$S_n = 1 + X_1 + X_2 + \cdots + X_n$$ where $X_1,X_2, ...$ are i.i.d. ...
3
votes
5answers
554 views

Where did it come from? (derivative of exponential)

We all know this rule: $\text{If } y = a^{f(x)} \text{ then } y' = a^{f(x)} \: f'(x)\ln a$ In my book there is the example: Find $\frac{d}{dx}\left((x^{2} + 1)^{\sin x}\right)$ According to ...
3
votes
3answers
39 views

Value of sine of complex numbers

I stumbled upon a problem with evaluating the sine function for complex arguments. I know that in general I can use $$ \sin(ix)=\frac{1}{2i}(\exp(-x)-\exp(x))=i\sinh(x). $$ But I could also write the ...
0
votes
1answer
33 views

Solving Laplace Transform of $-e^{-at}u(-t)$

I have found a problem in applying Laplace Transform to $-e^{-at}u(-t)$ I am doing these steps: $$ = - \int_{-\infty}^{+\infty} e^{-at}u(-t) e^{-st}dt$$ $$ = - \int_{-\infty}^{0} e^{-at} ...
1
vote
0answers
13 views

exponentialfunction poofs with generating functions

I have two things to proof. $(e^{ax})^{-1}=e^{-ax}$ and $(e^{ax})^{m}=e^{(ma)x}$ I know the power series of $e^x$ and $e^{ax}$ and that $e^{ax}e^{bx}=e^{(a+b)x}$ I tried it forward and backward: ...
0
votes
1answer
16 views

Bounding the growth of f(u)

Given a function $f(u) \le 2{\sqrt u}\,f({\sqrt u}) + 1$. We need to bound the growth of $f(u)$. If we expand this by recursively substituting, we get some series like $1 + \left\{ ...