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
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 :
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
= (—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) —
Show that /
is differentiable and f'(z) =
(iv) Same as part (iii), but for negative integers n, with the restriction z ^ 0.