Complex Logarithm - Branches of The Complex Logarithm

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 L(z) that is continuous on all of ? Unfortunately, the answer is no. To see why, imagine tracking such a logarithm function along the unit circle, by evaluating L at eiθ as θ increases from 0 to 2π. For simplicity, suppose that the starting value L(1) is 0. Then for L(z) to be continuous, L(eiθ) must agree with iθ as θ increases (the difference is a continuous function of θ taking values in the discrete set ). In particular, L(e2πi) = 2πi, but e2πi = 1, so this contradicts L(1) = 0.

To obtain a continuous logarithm defined on complex numbers, it is hence necessary to restrict the domain to a smaller subset U 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, U should be an open set. Also, it is reasonable to assume that U is connected, since otherwise the function on different components of U would be unrelated to each other. All this motivates the following definition:

A branch of log z is a continuous function L(z) defined on a connected open subset U of the complex plane such that L(z) is a logarithm of z for each z in U.

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

\log(1+u)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} u^n = u - \frac{u^2}{2} + \frac{u^3}{3} - \cdots \,

converges locally uniformly for |u| < 1, so setting z = 1+u defines a branch of log z on the open disk of radius 1 centered at 1. (Actually, this is just a restriction of Log z, as can be shown by differentiating the difference and comparing values at 1.)

Once a branch is fixed, it may be denoted "log z" 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 "log z" to have a precise unambiguous meaning.

Read more about this topic:  Complex Logarithm

Famous quotes containing the words branches of the, branches of, branches and/or complex:

    “In the woods in a winter afternoon one will see as readily the origin of the stained glass window, with which Gothic cathedrals are adorned, in the colors of the western sky seen through the bare and crossing branches of the forest.”
    —Ralph Waldo Emerson (1803–1882)

    ““I couldn’t afford to learn it,” said the Mock Turtle with a sigh. “I only took the regular course.”
    “What was that?” inquired Alice.
    “Reeling and Writhing, of course, to begin with,” the Mock Turtle replied; “and then the different branches of Arithmetic—Ambition, Distraction, Uglification, and Derision.”
    “I never heard of ‘Uglification,’” Alice ventured to say.”
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)

    “We are nothing but ceremony; ceremony carries us away, and we leave the substance of things; we hang on to the branches and abandon the trunk and body.”
    —Michel de Montaigne (1533–1592)

    “The human mind is so complex and things are so tangled up with each other that, to explain a blade of straw, one would have to take to pieces an entire universe.... A definition is a sack of flour compressed into a thimble.”
    —Rémy De Gourmont (1858–1915)