Complex Logarithm - Definition of Principal Value

Definition of Principal Value

For each nonzero complex number z, the principal value Log z is the logarithm whose imaginary part lies in the interval (−π,π]. The expression Log 0 is left undefined since there is no complex number w satisfying ew = 0.

The principal value can be described also in a few other ways.

To give a formula for Log z, begin by expressing z in polar form, z = reiθ. Given z, the polar form is not quite unique, because of the possibility of adding an integer multiple of 2π to θ, but it can be made unique by requiring θ to lie in the interval (−π,π]; this θ is called the principal value of the argument, and is sometimes written Arg z. Then the principal value of the logarithm can be defined by

For example, Log(-3i) = ln 3 − πi/2.

Another way to describe Log z is as the inverse of a restriction of the complex exponential function, as in the previous section. The horizontal strip S consisting of complex numbers w = x+yi such that −π < y ≤ π is an example of a region not containing any two numbers differing by an integer multiple of 2πi, so the restriction of the exponential function to S has an inverse. In fact, the exponential function maps S bijectively to the punctured complex plane, and the inverse of this restriction is . The conformal mapping section below explains the geometric properties of this map in more detail.

When the notation log z appears without any particular logarithm having been specified, it is generally best to assume that the principal value is intended. In particular, this gives a value consistent with the real value of ln z when z is a positive real number. The capitalization in the notation Log is used by some authors to distinguish the principal value from other logarithms of z.

A common source of errors in dealing with complex logarithms is to assume that identities satisfied by ln extend to complex numbers. It is true that eLog z = z for all z ≠ 0 (this is what it means for Log z to be a logarithm of z), but the identity Log ez = z fails for z outside the strip S. For this reason, one cannot always apply Log to both sides of an identity ez = ew to deduce z = w. Also, the identity Log(z₁z₂) = Log z₁ + Log z₂ can fail: the two sides can differ by an integer multiple of 2πi : for instance,

The function Log z is discontinuous at each negative real number, but continuous everywhere else in . To explain the discontinuity, consider what happens to Arg z as z approaches a negative real number a. If z approaches a from above, then Arg z approaches π, which is also the value of Arg a itself. But if z approaches a from below, then Arg z approaches −π. So Arg z "jumps" by 2π as z crosses the negative real axis, and similarly Log z jumps by 2πi.