the definition of on as the inverse of the restriction of to the interval : there are infinitely many real numbers with , 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. Branches have the advantage that they can be evaluated at complex numbers. On the other hand, the function on the Riemann surface is elegant in that it packages together all branches of the logarithm and does not require an arbitrary choice as part of its definition. Principal value Definition For each nonzero complex number , the principal value is the logarithm whose imaginary part lies in the interval . The expression is left undefined since there is no complex number satisfying . When the notation 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 when is a positive real number. The capitalization in the notation is used by some authors to distinguish the principal value from other logarithms of . Calculating the principal value The polar form of a nonzero complex number is , where is the absolute value of , and is its argument. The absolute value is real and positive. The argument is defined up to addition of an integer multiple of . Its principal value is the value that belongs to the interval , which is expressed as . This leads to the following formula for the principal value of the complex logarithm: For example, , and . The principal value as an inverse function Another way to describe is as the inverse of a restriction of the complex exponential function, as in the previous section. The horizontal strip consisting of complex numbers such that is an example of a region not containing any two numbers differing by an integer multiple of , so the restriction of the exponential function to has an inverse. In fact, the exponential function maps 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. The principal value as an