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ludwig1

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In summary, 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 different from singularities in individual analytic functions as it involves the interaction between multiple functions. Studying these singularities is important in understanding complex functions and their behavior. They can be classified into several types and cannot be avoided in general, but can be avoided in certain cases by manipulating the functions involved.

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ludwig1

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S.G. Janssens

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I think the most staightforward (but perhaps not the most handy?) approach would be to just consider Laurent series for $f$ and $g$ onludwig said:

Assuming that "singularity" means either "pole" or "essential singularity", you could start by considering the case that $f$ and $g$ have poles of (finite) orders $n$ and $m$ in $\mathbb{N}$, respectively. Next, you could look at the case that one or both of $f$ and $g$ have essential singularities (i.e. infinitely many non-zero terms occur in the principal parts of their Laurent series).

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math771

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Yes, both f(z)^2 and f(z)g(z) will have singularities at z=0. This can be seen by considering the definition of a singularity - a point at which the function is not defined or becomes infinite.

For f(z)^2, we can see that at z=0, f(z)^2 will become undefined if f(0)=0, as any real number squared is still defined, but any complex number squared will have an imaginary component. Therefore, f(z)^2 will have a singularity at z=0.

For f(z)g(z), we can consider the case where f(z)=1/z and g(z)=z. Both of these functions have singularities at z=0, and when multiplied together, f(z)g(z)=1, which is defined at z=0. However, if we take the limit as z approaches 0, we can see that the function will become infinite, indicating a singularity at z=0.

In general, if both f(z) and g(z) have singularities at z=0, then any function formed by combining them (such as f(z)^2 or f(z)g(z)) will also have a singularity at z=0. This can be proven using the definition of a singularity and the properties of analytic functions.

Therefore, both f(z)^2 and f(z)g(z) will have a singularity at z=0.

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.

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