1
vote
1answer
38 views

How to find the Direct Discrete Laplace Transform of ${2n \choose n}$

Some time ago I developed a discrete version of the Laplace transform for the purpose of calculating sums and solve finite difference equations with constant coefficients. The notes below are a ...
2
votes
1answer
107 views

Functional form of a series of a product of Bessels

This question arises from my answer to an inverse Laplace transform question. The result I got took the form $$ f(t)= e^{-r_0 t/2} H(t-a) \left [ J_0\left(\frac{1}{2} a r_0\right) ...
2
votes
0answers
115 views

A rational integral with exponential denominator

Prove that: $$\int_{-\infty }^{+\infty }{\frac{{{x}^{4}}\text{d}x}{\left( \beta +{{\text{e}}^{x}} \right)\left( 1-{{\text{e}}^{-x}} \right)}}=\frac{\left( {{\pi }^{2}}+{{\ln }^{2}}\beta ...
0
votes
1answer
52 views

A improper integral on expontential

Evaluate: $$\int_{0}^{\infty }{\frac{\left( 1-{{\text{e}}^{-px}} \right)\left( 1-{{\text{e}}^{-qx}} \right)\left( 1-{{\text{e}}^{-rx}} \right)}{{{\text{e}}^{x}}}}\text{d}x,\ \ \ p>0,\ q>0,\ ...
0
votes
0answers
61 views

Which class of functions can be represented as $F(z)=\int A(t)z^t dt$?

If I have a holomorphic function $f(z)$, then I can write it as $$f(z)=\sum_{n=0}^\infty a_n z^n.$$ So these functions can be viewed as a generating function of the coefficients $\{a_n\}$ which have a ...
0
votes
2answers
137 views

Z-Transform Identity

I've come across an identity and would like to know if it has some sort of formal name or derivation or explanation or something! Also, I'm curious as to whether others are aware of such an identity. ...
2
votes
2answers
680 views

Calculate the Laplace transform

Help me calculate the Laplace transform of a geometric series. $$ f(t) = \sum_{n=0}^\infty(-1)^nu(t-n) $$ show that $$ \mathcal{L} \{f(t)\} = \frac{1}{s(1+\mathcal{e}^{-s})} $$ Edit: so far I ...
3
votes
1answer
663 views

Discrete Laplace Tranform.

Yesterday ago I was reading how the Laplace Transform can be interpreted as the continuous analog of the discrete functional dependance of the power series $$f(x) = \sum a(n) x^n$$ This is to say, ...