More of the Lattice The lattice extends indefinitely in all directions. If we understand what is happening in one parallelogram, for instance, where it has poles, we will know what is happening in all the other parallelograms in the lattice. So if this main parallelogram has a pole at p, all the other parallelograms also have a pole at the same position. Note that the lattice itself is only made up of the vertices of the parallelograms, but sometimes it is helpful to think about what is happening in the interior of the parallelograms (in other words, the points aω1 +bω2 where neither a nor b is in Z). The number of zeros of f over this period parallelogram is the same as the number of poles (each counted with multiplicity, which means that if we have multiple poles at the same point we count them separately) and is called the order of the function. We can show this is true by considering a cell, which is a period parallelogram shifted so that it has no poles or zeros on its boundary. If we take the integral 1 2πi ZC f0(z) f(z) dx around the boundary C of the cell, it will give us the difference between the number of zeros Channels • 2022 • Volume 6 • Number 2 Page 37
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