in . For example, the principal value defines a branch on the open set where it is continuous, which is the set obtained by removing 0 and all negative real numbers from the complex plane. Another example: The Mercator series converges locally uniformly for , so setting defines a branch of on the open disk of radius 1 centered at 1. (Actually, this is just a restriction of , as can be shown by differentiating the difference and comparing values at 1.) Once a branch is fixed, it may be denoted "" if no confusion can result. Different branches can give different values for the logarithm of a particular complex number, however, so a branch must be fixed in advance (or else the principal branch must be understood) in order for "" to have a precise unambiguous meaning. Branch cuts The argument above involving the unit circle generalizes to show that no branch of exists on an open set containing a closed curve that winds around 0. One says that " has a branch point at 0". To avoid containing closed curves winding around 0, is typically chosen as the complement of a ray or curve in the complex plane going from 0 (inclusive) to infinity in some direction. In this case, the curve is known as a branch cut. For example, the principal branch has a branch cut along the negative real axis. If the function is extended to be defined at a point of the branch cut, it will necessarily be discontinuous there; at best it will be continuous "on one side", like at a negative real number. The derivative of the complex logarithm Each branch of on an open set is the inverse of a restriction of the exponential function, namely the restriction to the image . Since the exponential function is holomorphic (that is, complex differentiable) with nonvanishing derivative, the complex analogue of the inverse function theorem applies. It shows that is holomorphic on , and for each in . Another way to prove this is to check the Cauchy–Riemann equations in polar coordinates. Constructing branches via integration The function for real can be constructed by the formula If the range of integration started at a positive number other than 1, the formula would have to be instead. In developing the analogue for the complex logarithm, there is an additional complication: the definition of the complex