For question involving exponential functions and questions on exponential growth or decay.
3
votes
1answer
35 views
Closed form for $n$-th derivative of exponential
I need the closed-form for the $n$-th derivative ($n\geq0 $):
$$\frac{\partial^n}{\partial x^n}\exp\left(-\frac{\pi^2a^2}{x}\right)$$
Thanks!
By following the suggestion of Hermite polynomials:
...
2
votes
1answer
39 views
Can the graph of $x^x$ have a real-valued plot below zero?
The function $f(x) = x^x$ gives a complex number only if x has an even denominator. I'm not sure about irrational numbers. Why, then, is the best graph I can find of that function that of Wolfram ...
0
votes
1answer
45 views
limit of an exponential function
I was trying to understand how we can approximate exp.
One example is:
$$ \exp(t) = \sum_{i=0}^\infty t^i/i! $$
however, why is the next true:
$$\lim_{x\to \infty}\exp \left ({\frac{t^2}{2!} ...
1
vote
1answer
56 views
Error bounds for $e$
Prove that for all $n\in\mathbb{N}_+$, we have $$(1+\frac{1}{n})^n>\sum_{k=0}^n \frac{1}{k!}-\frac{e}{2n}.$$
1
vote
0answers
33 views
Definite integration of a high order exponential function mixed with rational function
I would like to solve the integral
$$\int_{x>0} x e^{ax^m+bx^n}dx,\qquad m>n>0$$
11
votes
7answers
293 views
$\pi$ from the unit circle, $\sqrt 2$ from the unit square but what about $e$? [duplicate]
If one wants to introduce $\pi$ to a not mathematically savvy person, the unit circle would be a good choice. The unit square would be the way to go for $\sqrt 2$. But what about $e$? I've reviewed ...
2
votes
4answers
49 views
Find the point on the graph of $y=e^{2x}$ at which the tangent line passes through the origin
Find the point on the graph of $y=e^{2x}$ at which the tangent line passes through the origin.
Completely lost on this question, the wording is confusing here.
3
votes
1answer
89 views
Is $\pi$ to do with circles or power series?
To get straight to the point: is $\pi$ defined as the ratio of the circumference and diameter of a circle, or as the first non-zero root of the power series of $\sin{x}$?
If the former, then $\pi$ ...
1
vote
2answers
43 views
exponential function with values between 0 and 1 for x values between 0 and 1
I am looking for a function that fits well to be used as a weight with exponential behavior. My x values are between 0 and 1, and i would like this function to behave exponentially, so that only x ...
5
votes
1answer
80 views
Interesting definite integral involving exp and trig
I'm trying to evaluate the following integrals:
$$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \cos(\phi) d\phi$$
$$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \sin(\phi) d\phi$$
for which I want to find ...
1
vote
1answer
59 views
Logical explanation of Euler's formula
This question is a about (if not proving) at least guessing the Euler's formula.
I don't want the proof using the infinite sums.
We can guess by logic that for example that the equation ...
0
votes
2answers
39 views
Is it true that: $e^{\theta^{N}} = e^{N\theta} \:\: \forall N \in \mathbb{N}$?
I have a simple exponential power question about e (mathematical constant), is it true that:
$$e^{\theta^{N}} = e^{N\theta} \:\: \forall N \in \mathbb{N}$$
1
vote
1answer
27 views
A question about an expoential function
I got an exponential function as follows
$\displaystyle 1-\frac{1}{x}+\frac{e^{-x}}{x}$
Does anyone know how to approximate such a function in a simpler term? Many thanks!
3
votes
3answers
95 views
Definition of $\exp(x)$
I've been thought at school that the definition of $\exp(x)$ is the unique function satisfying $\exp'(x)=\exp(x)$ and $\exp(0)=1$. But I've never found this definition somewhere else. On Wikipedia ...
2
votes
1answer
60 views
Why is $e^{g(x)} = \pi$ where $g(x)$ is holomorphic in Weierstrass factorization of sine function?
Why is $e^{g(x)} = \pi$ where $g(x)$ is holomorphic in Weierstrass factorization of sine function? I just can't get why it's true.
2
votes
1answer
20 views
How to prove and derive coefficients that an exponential whose base decreases exponentially is a parabola
Suppose I have a function defined by this recurrence-relation:
$$R(0) = r$$
$$R(n) = R(n-1) * (1+G)d^{n-1}$$
Here r is a base value, G is a base growth rate (G>0) and d is a decay in the growth rate ...
0
votes
0answers
31 views
Differentiation of Rotation Matrix
I have some troubles on differentiating a time-varying rotational matrix $R(t) \in SO(3)$. If I use a axis-angle representation with the Rodrigues formula we have:
$R(t) = \exp(\,A(t) \theta(t)\,) = ...
3
votes
1answer
65 views
How to show limit definition of $e^{z}$ holds if $z \in \mathbb{C}$
It is well known that for $x\in \mathbb{R}$ we have
$$
e^{x} = \lim_{n \to \infty} \left(1+ \frac{x}{n}\right)^n.
$$
This follows quickly by considering logarithms and using L'Hospital's rule.
...
5
votes
3answers
220 views
Can $\displaystyle\lim_{h\to 0}\frac{b^h - 1}{h}$ be solved without circular reasoning?
In many places I have read that $$\lim_{h\to 0}\frac{b^h - 1}{h}$$ is by definition $\ln(b)$. Does that mean that this is unsolvable without using that fact or a related/derived one?
I can of course ...
0
votes
1answer
44 views
How do I create an equation that decelerates past a certain value?
Apologies for my lack of pure maths, I am a programmer!
I currently have an equation in code that states that if a number goes below a certain value (in my case, 0.7) then the difference is dampened:
...
1
vote
3answers
47 views
Exponential relationship issue
I read this relation and I am not sure why this is true, is it I can't see why it would be?
$$(e^{ -i\pi/2})^{ -ix}\approx ie^{-\pi x/2} $$ I get that $e^{i\pi/2}=-i$, but I can't see why this ...
0
votes
0answers
36 views
Inequality involving the exponential function
Is there an easy way to see that
$$
j^{-k}C^{2k} (2k)! e^{C|\lambda |} \le C_1 e^{C_1 |\lambda | - j^{1/2}/C_1}
$$
if we take
$$
k = \Big [ \frac{j^{1/2}}{2C} \Big ] + 1
$$
where $[\cdot ]$ denotes ...
0
votes
1answer
36 views
Commuting in Matrix Exponential
Let $A$, and $B$ be commuting $n\times n$ matrices, i.e. $A.B = B.A$.
Let
\begin{equation}
\exp(A) = \sum_{i=0}^\infty\frac{1}{i!} A^i
\end{equation}
show that $\exp(A+B) = \exp(A).\exp(B)$.
14
votes
10answers
940 views
Why isn't $\lim_{x\to\infty}(1+\frac{1}{x})^{x}= 1$?
Given $\lim_{x\to\infty}(1+\frac{1}{x})^{x}$, why can't you reduce it to $\lim_{x\to\infty}(1+0)^{x}$, making the result "$1$"? Obviously, it's wrong, as the true value is $e$. Is it because the ...
1
vote
1answer
52 views
Derive the PDF of the log-normal distribution?
If $X \sim N(0,1)$ and $Y = e^X$, find the PDF of $Y$ using the two methods:
(i) Find the CDF of of $Y$ and then differentiate. Use the notation $\Phi(x)$ and $\phi(x)$ for the CDF and PDF of $X$ ...
0
votes
2answers
40 views
Definition of logarithm in complex domain
My first question is:
What is the proper definition of logarithmic function $f(z)=\ln{z}$.
where $z\in \mathbb{C}$.
quoting Wikipedia.
a complex logarithm function is an "inverse" of the ...
4
votes
3answers
66 views
what if take limit to negative infinity in the definition of e as a limit
By definition $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = e.$$
But what about a similar limit where $n$ tends to negative infinity, i.e, $$\lim_{n\to -\infty}\left(1+\frac{1}{n}\right)^n?$$
...
0
votes
4answers
108 views
Plotting a sum of 2 exponentials looks like… well nothing
I have been trying to plot a sum of 2 exponentials where one has positive and the other a negative exponent. I tried to plot this function:
$$
f(x) = 30e^{2x} + 3e^{-2x}
$$
And i get nothing here ...
1
vote
1answer
59 views
How to find the inverse of an expotential
Please tell me how to find the solution for this.
$$y= 3 + x + e^{x}$$
The range or the domain is not given. I'm just asked to find the inverse function of this.
2
votes
2answers
58 views
A non-zero function satisfying $g(x+y)=g(x)g(y)$ must be exponential function
Let $g$ be a non-zero function satisfying $g(x+y)=g(x)g(y)$. Show that the function must be exponential function.
1
vote
1answer
38 views
newtons cooling problem
A cup of coffee with cooling constant k = .09 min^-1 is placed in a room at tempreture 20 degrees C.
How fast is the coffee cooling(in degrees per minute) when its tempreture is T = 80 Degrees C?
Is ...
0
votes
1answer
36 views
Exponential Functions Carbon Dating
The question is
a paleontologist discovers remains of animals that appear to have died at the onset of the Holocene ice age, between 10000 and 12000 years ago. what range of C^14 to C^12 ratio would ...
2
votes
3answers
61 views
Incoherence using Euler's formula
Using the relation $\ e^{ix} = \cos(x) + i\sin(x)$ and substituting for $\ x = \pi$, we have the well-known Euler identity, $ e^{i\pi} = -1$. Substitute also for $ x = -\pi $, we have $ e^{-i\pi} = ...
1
vote
1answer
43 views
Solving basic exponential equation with logs
I am having trouble with this grade 12 pre-calc question that I am sure will be elementary to most of you. I understand most of it but I do not understand one of the steps.
These are the steps in my ...
13
votes
11answers
940 views
Intuitive proofs that $\lim\limits_{n\to\infty}\left(1+\frac xn\right)^n=e^x$
At this link someone asked how to prove rigorously that
$$
\lim_{n\to\infty}\left(1+\frac xn\right)^n = e^x.
$$
What good intuitive arguments exist for this statement? Later ...
0
votes
1answer
45 views
Expected value of minimum of exponentials
I am not sure of the following. I have $(i-1)$ exponential random variables with rates $\theta$ and $\mu$ and I want the expected value that the particular $\mu$ random variable is the minimum. Think ...
0
votes
0answers
34 views
Finding the zeroes of finite exponential sum
I want to find the zeroes for functions that look like this (an example):
$f(t) = k_1e^\left(a_1t\right)+k_2e^\left(a_2t\right)+k_3e^\left(a_3t\right)$
Where all a are negative real numbers, so this ...
0
votes
1answer
111 views
calculus Population differential equation
Let P(t) be the population of a city after t years. In this city the death rate is alpha times the population size, the birth rate is beta times the population size and the immigration out of city is ...
2
votes
1answer
36 views
Evaluation of an integral involving hyperbolic sine and exponential
I am wondering if the following integral can be reduced to either a closed form involving elementary functions, or well-known special functions (such as $\operatorname{erf}$, Bessel functions, etc.):
...
0
votes
0answers
49 views
Definition of $\exp(A)$ in terms of spectral decomposition.
I am read this question Plugging a matrix multiplied by an imaginary number in the exponential function. Here the question'author defined $$\exp(A) := \sum_{1\le k\le n}\exp(\lambda_k) P_k$$
What ...
1
vote
1answer
22 views
Using EXP in equation
I have the following equation:
$\ y' = te^{-2t} - 2y$
Where e is the exponential function. However when I see this being used I see EXP(x) and I don't understand how i'd write the equation with ...
1
vote
2answers
101 views
Evaluating a double integral involving exponential of trigonometric functions
I am having trouble evaluating the following double integral:
$$\int\limits_0^\pi\int\limits_0^{2\pi}\exp\left[a\sin\theta\cos\psi+b\sin\theta\sin\psi+c\cos\theta\right]\sin\theta d\theta\, d\psi$$
...
1
vote
1answer
58 views
What is the indefinite integral of $\int e^{\frac{1}{x^2 - a^2}} dx$?
I am looking for a solution to the following integral and finding it quite hard to find one ($|x| < a$):
$$ \int e^{\frac{1}{x^2 - a^2}} dx $$
I've tried to solve it with several substitutions, ...
2
votes
1answer
112 views
Variance of $\exp(-x)$
Hi I have been struggling to find the variance of the $\exp(-x)$ in terms of $\exp$.
For the function Y = exp (-x) where X is N (0,1) show that the variance of Y = $\exp(\exp-1)$
This is what I ...
2
votes
2answers
88 views
What is $ \lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$?
How to solve the following limit question?
$$\lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$$
Thanks a lot.
7
votes
0answers
58 views
Identity for $e$ in terms of the Fibonacci sequence.
The following identity appears in Martin Gardner's paper, "Dr. Matrix on Little Known Fibonacci Curiosities:
$$e = \frac{1 + 1 + \frac{2}{2!} + \frac{3}{3!} + \frac{5}{4!} + \frac{8}{5!} + ...
2
votes
3answers
35 views
Comparison test integral convergence
$$\int_0^{\infty} \frac{e^x}{x^x} \,\mathrm dx$$
How can I tell if this integral converges or not? I was thinking of using the comparison test, but I can't think of anything to compare it to. Could ...
1
vote
1answer
32 views
Connected set on complex plane
What's the numebr of connected components for the set of complex numbers $\{e^z:|z|=1\}$ on the complex plane?
Remark: It represents a simple closed curve which intersects the real axis at points ...
1
vote
0answers
24 views
Inequality of Partial Taylor Series
For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds:
$$\sum_{k=0}^{N} \frac{x^k}{k!} ...
4
votes
6answers
251 views
About $\lim \left(1+\frac {x}{n}\right)^n$
I was wondering if it is possible to get a link to a rigorous proof that
$$\displaystyle \lim \left(1+\frac {x}{n}\right)^n=\exp x$$
