4
votes
1answer
139 views

What $\alpha$ such that if $xy=\alpha$, then $e^{-x}+e^{-y}\geq 2e^{-\sqrt \alpha} $?

For every $ x,y \gt 0$, if $ xy=\alpha$, then we have $$e^{-x}+e^{-y}\geq 2e^{-\sqrt \alpha} $$ What are the possible values of $\alpha$? $2 < e^{1/(n+1)} + e^{-1/n}$ led to this problem. ...
1
vote
1answer
26 views

Prove $e^c>c^e$ if $c>0$ and $e \neq c$ using graph.

I am on this question where it tells me to show $e^c>c^e$ if $c>0$ and $e \neq c$ using the graph of $\dfrac{(log(x))}{x}$. Now it is obvious that the graph reaches a maximum at $x=e$ but how ...
3
votes
1answer
68 views

On the equation $\exp(a x+b)=\ln(x)$

I am confronted with: $$\exp(a x+b)=\ln(x)$$ for $a,b$ reals and $a<0$, $b>0$. I need the (unique) solution for $x$. My first target is (if it exists) an analytic solution in terms of ...
2
votes
0answers
30 views

Showing the exponential and logarithmic functions are unique in satisfying their properties

The question asks to prove that there exists a unique function defined on $\Bbb R$ and satisfying the following conditions: 1) $f(1) = a$ $(a>0, a \neq 0)$ 2) $f(x_1) \cdot f(x_2) = f(x_1 + ...
0
votes
2answers
57 views

Calculate the integral $\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz$

I am looking to solve $$\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz,$$ where $\varGamma$ is the contour $|z|=4\pi/3$. We have been asked first to consider $e^{z}=1$ and $e^{z}=-1$ which I get to be ...
0
votes
3answers
125 views

Solve these systems of equations

Consider the two equations below: $$ y_{1}=\left(1-\frac{a_{1}}{x}\right)e^{-\dfrac{\alpha\, a_{1}}{x}}\\ y_{2}=\left(1-\frac{a_{2}}{x}\right)e^{-\dfrac{\alpha\, a_{2}}{x}} $$ Given $y_{1}$, ...
0
votes
2answers
80 views

Derivative of $\exp(x)$

we defined $\exp(x) = \sum_{n=0}^\infty \frac{x^n}{n!}$ in the proof of the derivative one book starts to take the derivative of each single term i.e. $\dfrac{d(1 + x + x^2/2! +...)}{dx}$ and the ...
-1
votes
5answers
101 views

Prove $\exp(x) \geq 1+x \forall x \in \mathbb{R}$

I've managed to prove the statement for $x \geq 0 $ and $x \leq -1$ but I can't manage to construct a proof for $ -1 < x < 0 $ My lecture done it by proving $ \exp(x) - (1+x) = \displaystyle ...
4
votes
1answer
106 views

Proving continuity of exp(x)

Well, my teacher went through a method of proving continuity of $\exp(x)$ which I don't like, so I tried to go about it a different way: We have proved the following (which I use) $\exp(x+y) = ...
0
votes
1answer
85 views

Occurrence of $e$ in intersecting circles.

Consider two identical circles that share a radius such that they intersect. The radii of the circles are $\pi\over 2$. If this new shape sits such that its major axis is horizontal and the shortest ...
0
votes
1answer
91 views

Commuting Exponential Matrices

Let $x(t)=\exp(tA)\exp(tB)$ and $y(t)=\exp(t(A+B))$. Show that if $AB=BA$ then $x(t)$ and $y(t)$ satisfy the same initial value problem for ODEs and therefore must be equal. $A, B$ square matrices. ...
5
votes
9answers
2k views

Proving $\lim \limits_{n\to +\infty } \left(1+\frac{x}{n}\right)^n=\text{e}^x$.

I knew that $e^x=\lim \limits_{n\to+\infty }{\left(1+\frac{x}{n}\right)^n}$. But I've never seen its proof. So I tried to prove it using $\exp(\ln x)=\ln(\exp(x))=x$. Here is what I've tried so far : ...
0
votes
2answers
63 views

Derivative of $-e^y = 0$?

I stumbled upon this on wolfram alpha and still wonder why $-e^x$ equals $0$ (third step).
2
votes
1answer
404 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 $\exp(A+B) = \exp(A).\exp(B)$.
1
vote
2answers
102 views

How to show the double factorial isn't a polynomial

$(2n-1)!! = \dfrac{(2n)!}{2^{n} \times n!}$ I was wondering how you prove the double factorial is exponential. I guess you have to prove that for all $m$ and $\alpha$ that there exists an $n$ such ...
10
votes
2answers
202 views

Prove that $(e+x)^{e-x}>(e-x)^{e+x}$

I get stuck with proving that $$(e+x)^{e-x}>(e-x)^{e+x}$$ for $x \in (0, e)$. All I know, is that it is doable with Jensen inequality, and I started with defining $$f(x)=(e+x)^{e-x}$$ and further ...
5
votes
3answers
98 views

Stuck on this integral involving exp and the floor function

Here is the integral $$\int_0^\infty \lfloor x \rfloor e^{-x}dx$$ Here is what I have so far: $$I = \sum_{n=0}^\infty \int_n^{n+1} n e^{-x}dx$$ $$ = \sum_{n=0}^\infty -ne^{-n-1} + ne^{-n}$$ $$ = ...
0
votes
1answer
603 views

Prove that the exponential function is sequentially continuous?

I am supposed to use the definition of the exponential function to prove that if x is a real number and the modulus of x is less than 1, the modulus of exp(x)-1 is less than or equal to (e-1)*modulus ...
1
vote
1answer
437 views

Use Cauchy's Multiplication Theorem and the Binomial Theorem to prove $\exp(x+y)=\exp(x)\exp(y)$

I am to use Cauchy's Multiplication Theorem and the Binomial Theorem in order to prove $\exp(x+y)=\exp(x)\exp(y) $ but I have no idea where to begin. All I can think of doing is setting $\exp(x)$ ...
2
votes
1answer
152 views

A functional equation related to the exponential function

Suppose that $f:\mathbb{R}\rightarrow (0,\infty)$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ are two functions and satisfies the following relation \begin{equation} f(xy)=f(y)^{g(x)} \end{equation} ...
2
votes
3answers
125 views

Find an exponential function with given condition

How can I have an example of an exponential function defined in the X range 1 - infinity, with values starting at 40 and converging to 1?
1
vote
2answers
338 views

Power Series Expansion Problem Analysis

Please show that for all $x,y\in\mathbb{R}$, $$e^{x+y} - e^xe^y = \lim_{k\to\infty} \sum_{n=1}^k \sum_{j = 0}^n\left(\frac{x^{k+j}}{(k+j)!}\frac{y^{n-j}}{(n-j)!} + ...