Let f(z) be analytic at z0 and f(z0) ≠ 0. Prove log f(z) does not

1.	Let f(z) be analytic at z0 and f(z0) ≠ 0. Prove log f(z) does not have a branch point at z0.
Answer» If f(z₀) ≠ 0, f(z) has no zero in a sufficiently small neighborhood of z₀. Therefore, the argument of f(z) does not change as we go around a closed contour in this neighborhood. As a result, the value of log f(z) does not change as we go around this closed contour. Therefore, z₀ is not a branch point of f(z).

Let f(z) be analytic at z0 and f(z0) ≠ 0. Prove log f(z) does not have a branch point at z0.

Answer»

If f(z₀) ≠ 0, f(z) has no zero in a sufficiently small neighborhood of z₀.

Therefore, the argument of f(z) does not change as we go around a closed contour in this neighborhood. As a result, the value of log f(z) does not change as we go around this closed contour. Therefore, z₀ is not a branch point of f(z).

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Let f(z) be analytic at z0 and f(z0) ≠ 0. Prove log f(z) does not have a branch point at z0.

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