6 0. Differentiability and the Cauchy-Riemann Equations

Proof. We recall from advanced calculus that continuity of the partial

derivatives at z implies that / is real-differentiable at z. •

Note. We remind the reader that in the rest of the book the word "differ-

entiable" will mean "complex-differentiable".

Some of the following exercises ask you to prove things that were stated

above. The results of several of the exercises will be used later without

comment.

Exercises

0.1. Suppose that / is defined in a neighboorhood of the complex number

z. Show that / is complex-differentiable at z if and only if there exists a

complex number a such that f(z + h) = f(z) + ah + o(h) as h — • 0.

0.2. Suppose that / is complex-differentiable at z. Show that / is continu-

ous at z.

0.3. Suppose that T : C — C is R-linear. Show that T is C-linear if and

only if T(iz) = iTz for all z.

0.4. Suppose that T :

M2

—

R2

is the M-linear mapping defined by the

matrix , (that is, T(x,y) = (ax + by, ex + dy)). Show that T is

C-linear if and only if a = d and b — —c.

Hint: Use the fact that i(x,y) = (—yyx). (If

ui(x,y)

= (—y,#)" makes

no sense to you then you should read Appendix 1 first.)

0.5. Suppose that / and g are complex-differentiable at z.

(i) Show that f + g and fg are differentiable at z (and that their derivatives

are given by the same formulas as in elementary calculus).

(ii) Suppose in addition that g(z) ^ 0 and show that f/g is differentiable at

z (with derivative given by the same formula as in elementary calculus).

(iii) Suppose that n is a positive integer and define f(z) —

zn.

Show that /

is differentiable and f'(z) =

nzn~x.

(iv) Same as part (iii), but for negative integers n, with the restriction z ^ 0.