Complex logarithm

In complex analysis, a complex logarithm of the non-zero complex number z, denoted by w = log z, is defined to be any complex number w for which e w = z. This construction is analogous to the real logarithm function ln, which is the inverse of the real exponential function e y, satisfying e lnx = x for positive real numbers x. In complex analysis, a complex logarithm of the non-zero complex number z, denoted by w = log z, is defined to be any complex number w for which e w = z. This construction is analogous to the real logarithm function ln, which is the inverse of the real exponential function e y, satisfying e lnx = x for positive real numbers x. Since any complex number has infinitely many complex logarithms (see next paragraph), the complex logarithm cannot be defined to be a function on the complex numbers, but only as a multivalued function. Settings for a formal treatment of this are, among others, the associated Riemann surface, branches, or partial inverses of the complex exponential function. If z is given in polar form as z = r⋅e i⋅θ (r and θ real numbers with r > 0), then w0 = ln(r) + i⋅θ is one logarithm of z. Since z = r⋅e i⋅(θ + 2kπ) exactly for all integer k, adding integer multiples of 2π to the argument θ gives all the numbers that are logarithms of z: All these complex logarithms of z are on a vertical line in the complex plane at the distance ln(r) to the origin. Since every nonzero complex number z has infinitely many logarithms, care is required to give such a notion an unambiguous meaning. Sometimes the notation ln –instead of log– is used when addressing the complex logarithm. For a function to have an inverse, it must map distinct values to distinct values, that is, it must be injective. But the complex exponential function is not injective, because ew+2πi = ew for any w, since adding iθ to w has the effect of rotating ew counterclockwise θ radians. So the points equally spaced along a vertical line, are all mapped to the same number by the exponential function. This means that the exponential function does not have an inverse function in the standard sense. There are two solutions to this problem. One is to restrict the domain of the exponential function to a region that does not contain any two numbers differing by an integer multiple of 2πi: this leads naturally to the definition of branches of log z, which are certain functions that single out one logarithm of each number in their domains. This is analogous to the definition of arcsin x on as the inverse of the restriction of sin θ to the interval : there are infinitely many real numbers θ with sin θ = x, but one arbitrarily chooses the one in . Another way to resolve the indeterminacy is to view the logarithm as a function whose domain is not a region in the complex plane, but a Riemann surface that covers the punctured complex plane in an infinite-to-1 way.