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ludwig1
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Suppose f,g:ℂ→ℂ are analytic with singularities at z=0. I was wondering whether f(z)^2 or f(z)g(z) will have a singularity at z=0? For each, can you give me a proof or a counterexample?
I think the most staightforward (but perhaps not the most handy?) approach would be to just consider Laurent series for $f$ and $g$ on the same annulus centered at the origin. What do you know about the coefficients in the product of two such series?ludwig said:Suppose f,g:ℂ→ℂ are analytic with singularities at z=0. I was wondering whether f(z)^2 or f(z)g(z) will have a singularity at z=0? For each, can you give me a proof or a counterexample?
A singularity in the product of analytic functions refers to a point where the product of two or more analytic functions becomes undefined or infinite. It is a point where the function is not well-defined or smooth.
Singularities in the product of analytic functions are different from singularities in individual analytic functions because they involve the interaction between two or more functions. In individual analytic functions, a singularity may occur at a point where the function is not defined or has a pole. In the product of analytic functions, a singularity can occur at a point where both functions have singularities, or where the functions have a common zero.
Studying singularities in the product of analytic functions is important in understanding the behavior of complex functions and their properties. In particular, it helps us understand the behavior of functions near singularities and their impact on the overall behavior of the function. This knowledge is crucial in various fields such as engineering, physics, and mathematics.
Singularities in the product of analytic functions can be classified into several types, including removable, poles, essential, and branch points. Removable singularities occur when the function can be extended to the singularity point. Poles are singularities where the function goes to infinity. Essential singularities are points where the function has an infinite number of values. Branch points are singularities where the function has multiple values, and the choice of value depends on the path taken to approach the point.
In general, singularities in the product of analytic functions cannot be avoided since they are inherent properties of the functions involved. However, in some cases, it is possible to manipulate the functions to avoid certain types of singularities. For example, avoiding zeros or poles of a function can help avoid singularities in the product of analytic functions involving that function.