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70 2. Evaluation, Division, and Expansion

quantity xfz + yfY. What does this equal when deg f = l? deg f = 2?

Make a conjecture and prove it. Does the property that you have found

characterize homogeneous polynomials (i.e. if a polynomial has the prop-

erty, must it be homogeneous)? What is the generalization for more than

two variables?

E.25. Cauchy-Riemann Conditions. Consider a polynomial f (2) over

C of the complex variable z. If we make the substitution z = x + yi, and

separate out the real and imaginary parts, we can express f(z) in the form

u(x, Y) + iv(x, Y), where ‘u and v are polynomials in R[t, y]. For example,

if f(z) = 3z2 + (2 + i)z - (2 - 3i), verify that

f (x + Yi> = 3(x2 + 2xyi - y2) + (2 + i)(x + yi) - (2 - 3i)

= [3(x2-y2)+2x-y-22]+[6xy+x+2y+3]i,

so that, in this case U(X, y) = 3(x2 - y2) + 2x - y - 2

Thus, each complex polynomial f (2) corresponds to two real polynomials
~(2, y), v(z) y). What pairs {u, v} of polynomials arise in this way? Is it
possible to find a corresponding f for any given pair, or must there be some
relation connecting u and v?
To look at a simple example, show that it is not possible to find a complex
polynomial f( z ) f or which ~(2, y) = z and v(x, y) = 0. (Such a polynomial
would have to satisfy f(x + iy) = z for all real x and y.) If a(~:, y) = z,
what are the possibilities for v( 2, y)?
It turns out that there are two simple equations connecting the partial
derivatives u2, v, , uY, vY. By looking at the above example, as well as other
polynomials of low degree, make a conjecture. Now prove it, noting that
essentially you have to check your conjecture for the polynomial Z” for each
positive integer 12.
Compute the second order derivatives u,,, uYy, v,,, vyY for the above
example, as well as for other polynomials. Look for patterns and make
conjectures.
Suppose that you are given a polynomial u(z, y) in R[x, y]. Investigate
whether it is always possible to find a polynomial f(z) in C[Z] such that
f(x+yi) = u(x,Y)+qx,Y), f or some real polynomial v(x) y). For example,
you might look at ~(2, y) = 0, xy or x2. If such a polynomial v(x, y) exists,
how many possibilities are there?
The relationship connecting the first order partial derivatives of u and v is
not just a matter of idle curiosity. The natural generalization of polynomials
is a class of functions f(r) defined for a complex variable z which can be
expressed by an infinite series of monomials involving powers of z. The
real and imaginary parts of the functions in this class can be characterized
by the Cauchy-Riemann Conditions (in which the notion of derivative is