Logarithms and Powers

arguments and Arguments

If z ≠ 0, z has the polar coordinate representation

z = r eiθ, r = |z| > 0, θ = arg(z).

The angle θ = arg(z) is determined only modulo 2π.

If for every z ≠ 0 we make a particular choice Arg(z), then Arg(z) is called a Principle Argument function. For such a Principal Argument function, Arg(z) is continuous at z = z₀ iff for all z near z₀,

|Arg(z) − Arg(z0)| < π.

Common choices for Arg(z) are

Arg(z) = Arg[0,2π)(z) =arg(z), 0 ≤ arg(z) < 2 π, Arg(z) = Arg(−π,pi](z) = arg(z), −π < arg(z) ≤ π.

The first choice gives a Principal Argument Function which is continuous everywhere except along the positive x-axis . The second choice gives a Principal Argument Function which is continuous everywhere except along the negative x-axis .

logarithms and Logarithms

Definition. If z is a nonzero complex number then a logarithm of z is any complex number w such that

exp(w) = z.

Any logarithm is of the form

log(z) = ln|z| + i arg(z).

If Arg(z) is a Principal Argument Function, the function

Log(z) = ln|z|+ i Arg(z)

is called a Principal Branch of the Logarithm.

Exercise. Choose

Arg(z) = Arg(−π, π](z) = arg(z), − π < arg(z) ≤ π.

Show that

Log(z) ≡ ln|z| + i Arg(z)

is analytic except along the negative x-axis .

Hint: For ℜz > 0, draw a picture to show that Arg(x + i y) = arctan([y/x]) and verify the Cauchy-Riemann equations. Then give similar representations for Arg(z) in the regions ℑz > 0 and ℑz < 0.

powers and Powers

If z ≠ 0 and a is any complex number , z^a is any complex number of the form

za = ea log(z) = ea ·(ln|z| + i arg(z)).

In general z^a has more than one possible value. Choosing a Principle Argument Function Arg(z) give a Principal Branch of the Power Function, which is analytic at the places where the corresponding Principal Branch Logarithm Function Log(z) is analytic (and perhaps elsewhere in special cases). The corresponding function z^a ≡ e^{a Log(z)} is called a branch of the power function z^a.

Note that at any point where Arg(z) is continuous, Log(z) and the Branch f(z) ≡ e^{a Log(z)} of the power function z^a are analytic, and

dza dz =a za−1    (same branches).

Exercises

1.

Show that iⁱ is real and find the values of all of its branches.

2.

For z ≠ 0, how many values are there for z^[1/2]?

3.

Show that for any branch, lim_{z → 0} z^[1/2] = 0.

If f(z) = e^[1/2]Log(z), is continuous and analytic at z, then

f′(z) = 1 2 f(z) .

4.

Show that it is NOT possible to define Arg(z) in such a way that

f(z) = z[1/2] = e[1/2]Log(z), = e[1/2]((ln|z| + i Arg(z)))

is analytic in the region {z| 0 < |z| < R}.

Choices of a Principal Argument Function

If D is a simply connected region not containing z=0, the function f(ζ) = [1/(ζ)] is analytic in D and for any path C_{z0 → z} in D from z₀ to z,

⌠ ⌡ z z0  1 ζ   dζ ≡ ⌠ ⌡ Cz0 → z  1 ζ   dζ,

the integral being independent of the particular path chosen. Fixing z₀ ∈ D and a fixed choice for Arg(z₀), we can define on D:

Log(z0) = ln|z0| + i Arg(z0), Log(z) = Log(z0) + ⌠ ⌡ z z0  1 ζ  dζ = ln|z| + i Arg(z), Arg(z) = Arg(z0) +ℑ ⌠ ⌡ z z0  1 ζ  dζ.

A simply connected region D which does not contain z=0 can be constructed as the complement of a branch cut which consists of any simple curve Cwhich has 0 as an initial point and extends to z=∞.

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File translated from T_EX by T_TH, version 4.04.
On 03 Aug 2014, 20:27.