| 释义 |
complex logarithm The complex logarithm is a multifunction. If w is a particular logarithm of z ≠ 0, meaning expw = z, then every value of logz has the form w + 2nπi where n is an integer. To define logz as a function, principal values can be chosen via a branch of the logarithm on a cut plane; for example, every z in the complement of [0,∞) can be uniquely written z = rexp(iθ), where r>0 and 0<θ<2π and we set logz = lnr + iθ. This defines a holomorphic function logz on the cut plane, with derivative 1/z but which has a discontinuity of 2πi across the cut. Branches of complex powers might then be defined by zα = exp(αlogz), which is again holomorphic and has derivative αzα–1.
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