0
votes
1answer
36 views

proving a z transform

I am having trouble demonstrating the Z transform of $a^{n-1}u(n-1)$ is $\frac{1}{(z-a)}$, as it says in this table. I try using the definition of the z transform, but it comes out different than ...
2
votes
0answers
83 views

Theta series and Riemann Hypothesis

in the paper http://www.fuchs-braun.com/media/dd209bf5c2203a87ffff80a3ffffffef.pdf section 2 ' Hilbert-Polya space' page: 180 the author introduce the Theta series $$ F(\phi(x))= ...
3
votes
0answers
98 views

Gram's series for integral equation

The prime counting function $ \pi(x) $ satisfies the integral equation $$ \log\zeta (s)= s\int_{0}^{\infty}dx \frac{ \pi (e^{t})}{e^{st}-1} \tag{0}$$ and it has the solution in terms of Gram's ...
0
votes
1answer
63 views

Help me to understand the Gaussian blurring (2)

Here is an unknown luminosity function $f(x,y)$ and its integration results: $$\begin{align*} p_{i,j} &= \frac{1}{\Delta_{i,j}}\iint\limits_{D_{i,j}} \! f(x,y) \, dx \, dy,\\ \Delta_{i,j} &= ...
0
votes
1answer
43 views

Help me to understand the Gaussian blurring

Here is an unknown luminosity function $f(x,y)$ and its integration results: $$p_{i,j}= \frac{\iint\limits_{D_{i,j}} \! f(x,y) \, dx \, dy}{\iint\limits_{D_{i,j}} \,dx\,dy}$$ I need to express the ...
26
votes
2answers
3k views

Why do engineers use the Z-transform and mathematicians use generating functions?

For a (complex valued) sequence $(a_n)_{n\in\mathbb{N}}$ there is the associated generating function $$ f(z) = \sum_{n=0}^\infty a_nz^n$$ and the $z$-Transform $$ Z(a)(z) = \sum_{n=0}^\infty ...
3
votes
1answer
118 views

Series around $s=1$ for an integral

Consider the function $$F(s)=\int_{1}^{\infty}\frac{\text{Li}(x)}{x^{s+1}}dx$$ where $\text{Li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral. What is the series expansion around $s=1$? ...