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2.9: Branch Cuts and Function Composition

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We often compose functions, i.e. f(g(z)). In general in this case we have the chain rule to compute the derivative. However we need to specify the domain for z where the function is analytic. And when branches and branch cuts are involved we need to take care.

Example 2.9.1

Let f(z)=ez2. Since ez and z2 are both entire functions, so is f(z)=ez2. The chain rule gives us

f(z)=ez2(2z).

Example 2.9.2

Let f(z)=ez and g(z)=1/z. f(z) is entire and g(z) is analytic everywhere but 0. So f(g(z)) is analytic except at 0 and

C - { 2πni, where n is any integer }

The quotient rule gives h(z)=ez/(ez1)2. A little more formally: h(z)=f(g(z)). where f(w)=1/w and w=g(z)=ez1. We know that g(z) is entire and f(w) is analytic everywhere except w=0. Therefore, f(g(z)) is analytic everywhere except where g(z)=0.

Example 2.9.3

It can happen that the derivative has a larger domain where it is analytic than the original function. The main example is f(z)=log(z). This is analytic on C minus a branch cut. However

ddzlog(z)=1z

is analytic on C - {0}. The converse can’t happen.

Example 2.9.4

Define a region where 1z is analytic.

Solution

Choosing the principal branch of argument, we have w is analytic on

C{x0,y=0}, (see figure below.).

So 1z is analytic except where w=1z is on the branch cut, i.e. where w=1z is real and 0. It's easy to see that

w=1z is real and 0 z is real and 1.

So 1z is analytic on the region (see figure below)

C{x1,y=0}

Note

A different branch choice for w would lead to a different region where 1z is analytic.

The figure below shows the domains with branch cuts for this example.

屏幕快照 2020-09-03 下午5.10.09.png

Example 2.9.5

Define a region where f(z)=1+ez is analytic.

Solution

Again, let's take w to be analytic on the region

C{x0,y=0}

So, f(z) is analytic except where 1+ez is real and 0. That is, except where ez is real and 1. Now, ez=exeiy is real only when y is a multiple of π. It is negative only when y is an odd mutltiple of π. It has magnitude greater than 1 only when x>0. Therefore f(z) is analytic on the region

C{x0,y=odd multiple of π}

The figure below shows the domains with branch cuts for this example.

屏幕快照 2020-09-03 下午5.14.51.png


This page titled 2.9: Branch Cuts and Function Composition is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.

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