the principal logarithm

If the logarithm has infinitely many values, we need a default, a standard pick so that Log z names one specific number. The principal logarithm is that default: it chooses the value whose imaginary part lands in the half-open interval from -pi to pi.

By definition Log z = ln|z| + i Arg z, where Arg z is the principal argument, the unique angle of z in the range -pi < Arg z <= pi. Capital L and capital A signal the principal choice. For example Log(1) = 0, Log(i) = i pi/2, Log(-1) = i pi (we keep +pi, the upper endpoint), and Log(1 + i) = ln(sqrt 2) + i pi/4 = (1/2) ln 2 + i pi/4. On the positive real axis Log z reduces to the ordinary real ln, so it extends the familiar logarithm. It is defined for every non-zero z.

Log z is holomorphic everywhere except on the cut along the negative real axis (including 0), and there its derivative is 1/z, just like the real case. The trade-off for being single-valued is a seam: Log z jumps by 2 pi i as you cross the negative real axis, because Arg z jumps from just below +pi to just above -pi there. So Log is continuous and differentiable on the slit plane, the plane with the negative real axis and origin removed, but not across the slit. Different textbooks and software sometimes cut along the negative axis with the value -pi instead of +pi, or cut elsewhere; always check which convention is in force before trusting a formula.