Complex Logarithms
The exponential function can be extended to a function which gives a complex number as ex for any arbitrary complex number x; simply use the infinite series with x complex. This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm. There are two difficulties involved: no x has ex = 0; and it turns out that e2πi = 1 = e0. Since the multiplicative property still works for the complex exponential function, ez = ez+2nπi, for all complex z and integers n.
So the logarithm cannot be defined for the whole complex plane, and even then it is multi-valued – any complex logarithm can be changed into an "equivalent" logarithm by adding any integer multiple of 2πi at will. The complex logarithm can only be single-valued on the cut plane. For example, ln i = 1/2 πi or 5/2 πi or −3/2 πi, etc.; and although i4 = 1, 4 log i can be defined as 2πi, or 10πi or −6 πi, and so on.
- Plots of the natural logarithm function on the complex plane (principal branch)
-
z = Re(ln(x+iy))
-
z = |Im(ln(x+iy))|
-
z = |ln(x+iy)|
-
Superposition of the previous 3 graphs
Read more about this topic: Natural Logarithm
Famous quotes containing the word complex:
“All propaganda or popularization involves a putting of the complex into the simple, but such a move is instantly deconstructive. For if the complex can be put into the simple, then it cannot be as complex as it seemed in the first place; and if the simple can be an adequate medium of such complexity, then it cannot after all be as simple as all that.”
—Terry Eagleton (b. 1943)