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1 - the residue at a removable singularity is always zero so by the residue theorem the integral around a closed simple contour is zero. Cauchy's theorem states the integral around a simple closed contour for an analytic function is zero so am I right in thinking that the integral being zero does not imply the function is analytic ? As it is also zero for functions with a removable singularity.

2 - I have seen the following example in a book where it is shown that (z-sinz)/z

^{3}has a removable singularity at z=0 but then it states that the series converges for all values of z but surely the function is undefined at z=0 ?

Thanks