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Physics LibreTexts

8.5: Exercises

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Exercise 8.5.1

Find the values of (i)i.

Answer

We can write i in polar coordinates as exp(iπ/2). Hence, (i)i=exp{iln[exp(iπ/2)]}=exp{i[iπ2+2πin]},nZ=exp[2π(n+14)],nZ.

Exercise 8.5.2

Prove that ln(z) has branch points at z=0 and z=.

Answer

Let z=rexp(iθ), where r>0. The values of the logarithm are ln(z)=ln(r)+i(θ+2πn),nZ. For each n, note that the first term is the real part and the second term is the imaginary part of a complex number wn. The logarithm in the first term can be taken to be the real logarithm.

For z0, we have r0 and hence ln(r). This implies that wn lies infinitely far to the left of the origin on the complex plane. Therefore, wn (referring to the complex infinity) regardless of the value of n. Likewise, for z, we have r and hence ln(r)+. This implies that wn lies infinitely far to the right of the origin on the complex plane, so wn regardless of the value of n. Therefore, 0 and are both branch points of the complex logarithm.

Exercise 8.5.3

For each of the following multi-valued functions, find all the possible function values, at the specified z:

  1. z1/3 at z=1.

  2. z3/5 at z=i.

  3. ln(z+i) at z=1.

  4. cos1(z) at z=i

Exercise 8.5.4

For the square root operation z1/2, choose a branch cut. Then show that both the branch functions f±(z) are analytic over all of C excluding the branch cut.

Exercise 8.5.5

Consider f(z)=ln(z+a)ln(za). For simplicity, let a be a positive real number. As discussed in Section 8.4, we can write this as f(z)=ln|z+aza|+i(θ+θ),θ±arg(z±a).

Suppose we represent the arguments as θ+(π,π) and θ(π,π). Explain why this implies a branch cut consisting of a straight line joining a with a. Using this representation, calculate the change in f(z) over an infinitesimal loop encircling z=a or z=a. Calculate also the change in f(z) over a loop of radius Ra encircling the origin (and thus enclosing both branch points).


This page titled 8.5: Exercises is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to the style and standards of the LibreTexts platform.

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