. Applications The complex logarithm is needed to define exponentiation in which the base is a complex number. Namely, if and are complex numbers with , one can use the principal value to define . One can also replace by other logarithms of to obtain other values of , differing by factors of the form . The expression has a single value if and only if is an integer. Because trigonometric functions can be expressed as rational functions of , the inverse trigonometric functions can be expressed in terms of complex logarithms. Since the mapping transforms circles centered at into vertical straight line segments, it is useful in engineering applications involving an annulus. Generalizations Logarithms to other bases Just as for real numbers, one can define for complex numbers and with the only caveat that its value depends on the choice of a branch of log defined at and (with ). For example, using the principal value gives Logarithms of holomorphic functions If f is a holomorphic function on a connected open subset of , then a branch of on is a continuous function on such that for all in . Such a function is necessarily holomorphic with for all in . If is a simply connected open subset of , and is a nowhere-vanishing holomorphic function on , then a branch of defined on can be constructed by choosing a starting point a in , choosing a logarithm of , and defining for each in . Notes References Analytic functions Logarithms