Re 1 z Figure 7.1.
1 Im z 1 Real and Imaginary Parts of f (z) = z .
Therefore, if f (z) is any complex function, we can write it as a complex combination f (z) = f (x + i y) = u(x, y) + i v(x, y), of two inter-related real harmonic functions: u(x, y) = Re f (z) and v(x, y) = Im f (z). Before delving into the many remarkable properties of complex functions, let us look at some of the most basic examples. In each case, the reader can directly check that the harmonic functions provided by the real and imaginary parts of the complex function are indeed solutions to the Laplace equation. Examples of Complex Functions (a) Harmonic Polynomials: As noted above, any complex polynomial is a linear combination, as in (7.2), of the basic complex monomials z n = (x + i y)n = un (x, y) + i vn (x, y). (7.7)
Their real and imaginary parts, un , vn , are the harmonic polynomials that we previously constructed by applying separation of variables to the polar coordinate form of the Laplace equation (4.94). The general formula can be found in (4.110). (b) Rational Functions: Ratios p(z) f (z) = (7.8) q(z) of complex polynomials provide a large variety of harmonic functions. The simplest case is x y 1 = 2 -i 2 . (7.9) 2 z x +y x + y2 Its real and imaginary parts are graphed in Figure 7.1. Note that these functions have an interesting singularity at the origin x = y = 0, but are harmonic everywhere else. 1/19/12 219
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Figure 7.2.
Real and Imaginary Parts of ez .
A slightly more complicated example is the function f (z) = z-1 . z+1 (7.10)
To write out (7.10) in real form, we multiply and divide by the complex conjugate of the denominator, leading to z-1 (z - 1)( z + 1) | z |2 + z - z - 1 x2 + y 2 - 1 2y = = = +i . 2 2 + y2 z+1 (z + 1)( z + 1) |z +1| (x + 1) (x + 1)2 + y 2 (7.11) Again, the real and imaginary parts are both harmonic functions away from the singularity x = -1, y = 0. Incidentally, the preceding manipulation can always be used to find the real and imaginary parts of general rational functions. (c) Complex Exponentials: Euler's formula f (z) = ez = ex cos y + i ex sin y (7.12)
for the complex exponential yields two important harmonic functions: ex cos y and ex sin y, which are graphed in Figure 7.2. More generally, writing out ec z for a complex constant c = a + i b produces the complex exponential function ec z = ea x-b y cos(b x + a y) + i ea x-b y sin(b x + a y), (7.13)
whose real imaginary and parts are harmonic functions for arbitrary a, b R. Some of these were found by applying the separation of variables method in Cartesian coordinates. (d) Complex Trigonometric Functions: The complex trigonometric functions are defined in terms of the complex exponential by adapting our earlier formulae (3.60): cos z = e i z + e- i z = cos x cosh y - i sin x sinh y, 2 e i z - e- i z = sin x cosh y + i cos x sinh y. sin z = 2i 220
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(7.14)
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Re (log z) = log | z | Figure 7.3.
Im (log z) = ph z Real and Imaginary Parts of log z.
The resulting harmonic functions are products of trigonometric and hyperbolic functions, and can all be written as linear combinations of the harmonic functions (7.13) derived from the complex exponential. Note that when z = x is real, so y = 0, these functions reduce to the usual real trigonometric functions cos x and sin x. (e) Complex Logarithm: In a similar fashion, the complex logarithm log z is a complex extension of the usual real natural (i.e., base e) logarithm. In terms of polar coordinates z = r e i , the complex logarithm has the form log z = log(r e i ) = log r + log e i = log r + i . Thus, the logarithm of a complex number has real part Re (log z) = log r = log | z | =
1 2
(7.15)
log(x2 + y 2 ),
which is a well-defined harmonic function on all of R 2 save for a logarithmic singularity at the origin x = y = 0. It is, up to multiple, the logarithmic potential (6.104) corresponding to a delta function forcing concentrated at the origin -- which played a key role in our construction of the Green's function for the Poisson equation. The imaginary part Im (log z) = = ph z of the complex logarithm is the phase (or argument) of z, also not defined at the origin x = y = 0. Moreover, the phase is a multiply-valued harmonic function elsewhere, since it is only specified up to integer multiples of 2 . Each nonzero complex number z = 0 has an infinite number of possible values for its phase, and hence an infinite number of possible complex logarithms log z, differing from each other by an integer multiple of 2 i , which reflects the fact that e2 i = 1. In particular, if z = x > 0 is real and positive, then log z = log x agrees with the real logarithm, provided we choose ph x = 0. Alternative choices append some integer multiple of 2 i , and so ordinary real, positive numbers x > 0 1/19/12 221
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