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I was studying for my Complex Analysis(an elementary undergraduate Complex Analysis course) exam and came across this question in the textbook. I know that in order for there to be a pole of some order, the limit of f(z) would have to go to infinity. I also know that if f(z) has a removable singularity then it is bounded. Thus, I am not sure how if two functions, each with poles of different orders, has a product with a removable singularity. Any advice, tips, and suggestions would be much welcomed. I have attached this question as an image. Please note, that this is for an elementary Complex Analysis course, so please nothing too advanced, I am really trying to understand the concepts. enter image description here

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Since $f$ has a zero of order $m$ at $z_0$, its Taylor series in the disk $|z - z_0 | < R$ looks like: $$ f(z) = a_m(z - z_0)^m + a_{m+1} (z - z_0)^{m+1} + a_{m+2}(z - z_0)^{m + 2} + \dots $$ Therefore, you can write $$ f(z) = (z - z_0)^m p(z),$$ where $$ p(z) = a_m + a_{m+1}(z - z_0) + a_{m+2}(z - z_0)^2 + \dots$$ Since $p(z)$ is given by a power series, and since all power series are holomorphic within their disk of convergence, it follows that $p(z)$ is holomorphic on the disk $| z - z_0 | < R$.

Similarly, since $g$ has a pole of order $l$ at $z_0$, its Laurent series on the punctured disk $0 < | z - z_0 | < R$ looks like: $$ g(z) = b_{-l}(z - z_0)^{-l} + b_{-l+1}(z - z_0)^{-l+1} + b_{-l+2}(z - z_0)^{-l+2} + \dots.$$ So you can write $$ g(z) = (z - z_0)^{-l}q(z),$$ where $$ q(z) = b_{-l} + b_{-l+1}(z - z_0) + b_{-l+2}(z - z_0)^2 + \dots$$ Since $q(z)$ is given as a power series, $q(z)$ is holomorphic on the disk $| z - z_0 | < R$; in particular, it is well-defined and holomorphic at the point $z = z_0$.

Finally, $$ f(z)g(z) = (z - z_0)^{m-l} p(z) q(z).$$ Since $m - l \geq 0$, and since $p(z)$ and $q(z)$ are both holomorphic on $| z - z_0 | < R$ (including at the point $z = z_0$), we see that the right-hand side is holomorphic on the disk $|z - z_0| < R$ (including at the point $z = z_0$). Thus we have succeeded in extending $f(z)g(z)$ to a holomorphic function on $| z - z_0 | < R$ (including at the point $z = z_0$), which proves that $f(z)g(z)$ has a removable singularity at $z = z_0$.

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