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

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1answer
48 views

Multiplicative inverse of the power series $e^x - c$ for $c \neq 1$.

We know that the power series $f(x)= e^x -c \in \mathbb C[[x]]$ for $c \neq 1$, has a multiplicative inverse, since it's constant coefficient is non-zero. I was wondering whether the inverse is known ...
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2answers
22 views

Trying to understand an expansion/limit from geometric sum to exponentials, what kind of rule is at play?

Can someone help me understand what's going on here? This is for a problem involving moment generating functions, which is related to statistics and probability, but I figured it was more of a math ...
2
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0answers
50 views

Does every even degree polynomial in the expansion of $\exp x$ have no real roots? [duplicate]

Does every even degree polynomial in the expansion of $\exp x$ have no real roots? I tried the first few values and it seems like it... Is this a known result?
2
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0answers
84 views

Tough integral with exp

Can anybody integrate this: $$ \int_0^K e^{i(A\sqrt{\mathstrut k^2+m^2} - Bk)} dk,$$ where $K$, $A$, $B$ and $m$ are real constants? Sorry folks, I didn't realise anybody would be interested in the ...
4
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2answers
152 views

Is $e^x=\exp(x)$ and why?

In the comments to this question a discussion came up wether we have $e^x=\exp(x)$ by definition and what the "correct" definition of $\exp(x)$ is. Building on that, I want to line out the problem ...
3
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4answers
60 views

Using the $\ln(\cdot)$ for $(1-e^{-x})$

The given function: $$B= A(1-(e^{-x}))$$ Now, I want to 'destroy' the e-function by taking the logarithm of it. First, since $\ln(ab) = \ln(a) + \ln(b)$ we get that $\ln(b) = \ln(a) + ...
2
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2answers
65 views

What is the fallacy of this proof?

I recently was working with square roots and came across this- $({\sqrt -1})$$=-1^\frac12$$=-1^\frac24$$=(-1^2)^\frac14$$=1^\frac14$$=1$ I understand that this is not true,but despite repeated ...
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0answers
26 views

How to solve the integral in this case?

If I have a Kernel from the path integral technique, and I want check if the product rule is valid in 2D, (I know it is) then I need to solve the following integral: \begin{equation} \text{Let} \quad ...
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1answer
18 views

exponential behavior from pattern of data

In the image below from this video lesson, the teacher shows how to get an exponential function from a pattern of data, also copied below. You can see that her solution using the formula (a)(b) to the ...
7
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5answers
382 views

Limit of a function involving a sequence.

I have the following problem: Suppose that $\lim_{n \to \infty} a_n = 0$. Prove that for any $x$ $$\lim_{n \to \infty} \left(1+a_n \frac{x}{n}\right)^n = 1.$$ I have tried replacing $a_n$ with ...
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1answer
47 views

If $(t-2)= e^{3(x-1)}$ then $x=?$ [closed]

If $(t-2)= e^{3(x-1)} $ then $x=?$. I guess I have to change the right side of the equation to get the x to the other side.
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2answers
140 views
0
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1answer
25 views

I want to find PDF by differentiating CDF and then from PDF, expected values of the following problem. .

Three light bulbs have independent exponentially distributed lifetimes with a common parameter $\lambda$. What is the probability distributed function and expected value of the time until the last ...
4
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2answers
98 views

What is intresting about $\sqrt{\log_x{\exp{\sqrt{\log_x{\exp{\sqrt{\log_x{\exp{\cdots}}}}}}}}}=\log_x{e}$?

Why does $$\sqrt{\log_x{\exp{\sqrt{\log_x{\exp{\sqrt{\log_x{\exp{\cdots}}}}}}}}}=\log_x{e}=\frac{1}{\ln{x}}$$ There only seems to be a relation when using square roots, but not for cubed roots or ...
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2answers
30 views

Exponential functions with negative base

Consider the function $f(x) = (-2)^x$, $x$ belongs to irrationals. For which $x$ does $f(x)$ belong to the reals.
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3answers
134 views

Simplifying integral:$\int_{0}^{\infty}{\exp\left(-\left(u^2+{ {\alpha^2}\over {16u^2t}}\right)\right)}~\mathrm{d}u$

$$I(t)=\int_{0}^{\infty}{\exp\left(-\left(u^2+{ {\alpha^2}\over {16u^2t}}\right)\right)}~\mathrm{d}u $$ where $\alpha$ and $t$ are positive constant. P.S.I would like to edit this problem, because ...
7
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4answers
460 views

Exponential Simultaneous Equations

Solve the following simultaneous equations: $$2^x + 2^y = 10$$ $$x + y = 4$$ Looking at it, it is obvious that the answers are $(3,1)$ and $(1,3)$, however, I was wondering if they could be solved ...
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2answers
40 views

How to calculate matrix exponential of a $2\times 2$ matrix with repeated e values

Specifically, I am trying to calculate the matrix exponential, $e^{At}$, where A = $\begin{bmatrix}-1 & 1\\-9 & 5\end{bmatrix}$. I calculated the the E values to be 2 with a multiplicity of 2 ...
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3answers
46 views

Express $\cosh 2x$ and $\sinh 2x$ in exponential form and hence solve for real values of $x$ the equation:$2 \cosh 2x - \sinh 2x =2$

Express $\cosh 2x$ and $\sinh 2x$ in exponential form and hence solve for real values of $x$ the equation: $2 \cosh 2x - \sinh 2x =2$ Here is my idea: $$2 \cosh 2x- \sinh 2x = ...
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1answer
32 views

try to create a more linear part in a sigmoid curve

I want to create a function for a curve which is similar to a sigmoid curve but the center part is linear. That curve must pass through 2 points (-1.5,20) and (1.5,80). The range for y value is ...
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2answers
37 views

How this exponential function converted into cosine function including phase shift?

In my text book I have found this two line in an example: $$ y(t)=\frac{2}{1+j}e^{jt}+\frac{2}{1-j}e^{-jt}-\frac{1}{1+j2}e^{j2t}-\frac{1}{1-j2}e^{-j2t}$$ ...
0
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1answer
25 views

Solve explicitly for time from a sum of exponentials

Suppose $f(t) = 0$. How can I solve for time, $t$, in the following expression. $f(t) = k_1{e}^{- \alpha t} + k_2{e}^{- \beta t} + k_3{e}^{- \gamma t}\left( k_4 sin(\omega_d t) + k_5 cos (\omega_d t) ...
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2answers
18 views

Approximation of a negative exponential model?

I am trying to get an approximation of this model, it is a negative exponential model introduced by Olson in 1963 "Energy storage and the balance of producers and decomposers in ecological systems". ...
2
votes
4answers
67 views

Derivative of $e^{\sin(\ln[2 \arctan(2x)])}$ [closed]

Let $$F(x) = e^{\sin(\ln[2 \arctan(2x)])}$$ and take the first derivative of $F(x)$. Could someone walk me through each step here? It seems to be a pretty straightforward chain rule problem but it ...
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3answers
43 views

Is it possible to find initial parameters when fitting triple exponential term function to data?

I'm trying to fit $f(x) = A \exp(Bx) + C \exp(Dx) + E \exp(F x) $ to data. I can finish off the fitting using Levenberg-Marquardt, but I'd like to find a quick way to calculate initial parameters. ...
2
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0answers
31 views

Bounds on the product of a matrix exponential and a vector

I have a control system with a state matrix $S = -B^{-1} A \in \mathbb{R}^{n\times n}$, where: $B$ is a strictly positive diagonal matrix $A$ is positive definite $M$-matrix I know that all the ...
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votes
2answers
46 views

Prove result of xy [closed]

If $$25^x = 7\quad \text{and}\quad 7^y = 125$$ then $xy=\frac{3}{2}$. Can someone explain me why $xy$ is equal to $\frac{3}{2}$? Thank you
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2answers
42 views

Largest domain where the function $e^z/(\sin z+\cos z)$ is analytic

I have a function $f(z) = \frac{\exp{z}}{\sin z+\cos z}$ and I need to show the region where $f(z)$ is analytic. My work so far :- As the function is the sum and product of holomorphic functions, I ...
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1answer
18 views

Unform convergence on a bounded subset of the complex plane

What does it mean when someone says that the exponential series converges uniformly on every bounded subset of the complex plane. I know the definition of convergence, uniform convergence of sequence ...
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2answers
30 views

Continuous Compound

You own an antique that is currently worth 1500, and whose value increases linearly at a rate of 175 per year. If the prevailing interest rate remains constant at 5%, per year compounded continuously, ...
2
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2answers
133 views

What is the solution of $a^b=a+b$ in terms of $a$?

Let $a, b$ be real numbers. Solve $$a^b=a+b$$ for $a$. If there isn't a solution with $a, b$ real, maybe $a, b$ should be complex. But no matter how hard I try, this is proving to be very ...
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1answer
43 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) = ...
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2answers
66 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 ...
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1answer
53 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
36 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 ...
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0answers
40 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
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1answer
45 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 ...
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1answer
39 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
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1answer
48 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
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1answer
92 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 ...
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1answer
37 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
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4answers
113 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 ...
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5answers
75 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. ...
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3answers
41 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 ...
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4answers
84 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 ...
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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 ...
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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 $\dfrac{a^{2/3}}{27^{1/3}}\cdot \dfrac{64^{2/3}}{a^{2/3}}$ ...
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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 $A(\varphi)=\left(\begin{matrix} \cos\varphi &\sin\varphi\\ -\sin\varphi & \cos\varphi\end{matrix}\right)$ . ...
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0answers
46 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 $n=-1$ ...
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 ...