8.5: Exercises
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Exercise 8.5.1
Find the values of (i)i.
- Answer
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We can write i in polar coordinates as exp(iπ/2). Hence, (i)i=exp{iln[exp(iπ/2)]}=exp{i[iπ2+2πin]},n∈Z=exp[−2π(n+14)],n∈Z.
Exercise 8.5.2
Prove that ln(z) has branch points at z=0 and z=∞.
- Answer
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Let z=rexp(iθ), where r>0. The values of the logarithm are ln(z)=ln(r)+i(θ+2πn),n∈Z. 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 z→0, we have r→0 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:
- z1/3 at z=1.
- z3/5 at z=i.
- ln(z+i) at z=1.
- cos−1(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(z−a). For simplicity, let a be a positive real number. As discussed in Section 8.4, we can write this as f(z)=ln|z+az−a|+i(θ+−θ−),θ±≡arg(z±a).