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19.4 The Complex Logarithm
19.4
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The Complex Logarithm
In real calculus, the natural logarithm is the inverse of the exponential function. For x > 0,
y
y = ln(x) if and only if x = e .
y
In this way, the real natural logarithm can be thought of as the solution of the equation x = e .
We will use this approach in developing a complex logarithm. Given z = 0, we want to solve for
w in the equation
w
e = z.
iθ
To do this, put z in polar form as z =re , and let w = u + iv to write
iθ
w
u iv
z =re = e = e e . (19.7)
iv
iθ
u
Since θ and v are real, |e |=|e |=1. Taking magnitudes in equation (19.7) gives us r =|z|=e ,
therefore
u = ln(r),
which is the real natural logarithm of the positive number r. But now equation (19.7) implies that
iv
iv
iθ
iθ
e = e . Then e /e = e i(v−θ) = 1. We know all the solutions of this equation, namely
i(v − θ) = 2nπi
for integer n. Then
v = θ + 2nπ.
iθ
In summary, given z = re with r = 0, there are infinitely many complex numbers w such that
w
e = z. All of these numbers are
w = ln(r) + iθ + 2nπi,
with n any integer. This leads us to define, for z = 0,
log(z) = ln(|z|) + iθ + 2nπi
in which θ is any argument of z and n can be any integer. The complex log is not a function in
the conventional sense because each nonzero z has infinitely many different complex logarithms.
EXAMPLE 19.13
√
Let z = 1 + i. In polar form, z = 2e i(π/4+2nπi) . The values of the logarithm of 1 + i are
√ π
log(1 + iz) = ln 2 + i + 2nπi .
4
In the complex plane, we can take the logarithm of a negative number.
EXAMPLE 19.14
Let z =−3. In polar form, z = 3e (π+2nπ)i = 3e (2n+1)πi . The values of the logarithm of −3are
log(−3) = ln(3) + (2n + 1)πi
in which n can be any integer.
October 15, 2010 18:5 THM/NEIL Page-689 27410_19_ch19_p667-694