analytic continuation On the region consisting of complex numbers that are not negative real numbers or 0, the function is the analytic continuation of the natural logarithm. The values on the negative real line can be obtained as limits of values at nearby complex numbers with positive imaginary parts. Properties Not all identities satisfied by extend to complex numbers. It is true that for all (this is what it means for to be a logarithm of ), but the identity fails for outside the strip . For this reason, one cannot always apply to both sides of an identity to deduce . Also, the identity can fail: the two sides can differ by an integer multiple of ; for instance, but The function is discontinuous at each negative real number, but continuous everywhere else in . To explain the discontinuity, consider what happens to as approaches a negative real number . If approaches from above, then approaches , which is also the value of itself. But if approaches from below, then approaches . So "jumps" by as crosses the negative real axis, and similarly jumps by . Branches of the complex logarithm Is there a different way to choose a logarithm of each nonzero complex number so as to make a function that is continuous on all of ? The answer is no. To see why, imagine tracking such a logarithm function along the unit circle, by evaluating as increases from to . If is continuous, then so is , but the latter is a difference of two logarithms of , so it takes values in the discrete set , so it is constant. In particular, , which contradicts . To obtain a continuous logarithm defined on complex numbers, it is hence necessary to restrict the domain to a smaller subset of the complex plane. Because one of the goals is to be able to differentiate the function, it is reasonable to assume that the function is defined on a neighborhood of each point of its domain; in other words, should be an open set. Also, it is reasonable to assume that is connected, since otherwise the function values on different components of could be unrelated to each other. All this motivates the following definition: A branch of is a continuous function defined on a connected open subset of the complex plane such that is a logarithm of for each