Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
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(i) there exists c ∈ C such that T(z) = cz for all z ∈ C;<br />
(ii) T is C-linear;<br />
(iii) T(i) = iT(1);<br />
(iv) the real 2×2 matrix representing T with respect to the standard basis <strong>of</strong> R2 may<br />
be written as<br />
� �<br />
a b<br />
A =<br />
−b a<br />
for some real a,b ∈ R.<br />
Pro<strong>of</strong>. (i) =⇒ (ii) =⇒ (iii) is obvious.<br />
(iii) =⇒ (i): Set c := T(1). For z = x+iy ∈ C, this means that<br />
T(x+iy) = T(x)+T(iy)<br />
= xT(1)+yT(i)<br />
= xT(1)+iyT(1)<br />
= zT(1)<br />
= cz.<br />
(iv) ⇐⇒ (iii): Let a,b,c,d ∈ R be such that<br />
A =<br />
� a b<br />
c d<br />
represents T with respect to the standard basis <strong>of</strong> R2 . Note that<br />
� �� � � �<br />
a b 1 a<br />
T(1) = = = a+ic,<br />
c d 0 c<br />
and<br />
Since<br />
we see that<br />
T(i) =<br />
� a b<br />
c d<br />
�� 0<br />
1<br />
�<br />
=<br />
�<br />
.<br />
� b<br />
d<br />
iT(1) = −c+ia,<br />
�<br />
= b+id.<br />
T(i) = iT(1) ⇐⇒ c = −b and d = a.<br />
Theorem 2.1 (Cauchy–Riemann Equations). Let D ⊂ C be open, and let z0 ∈ D.<br />
Let f: D → C and denote u := Ref, v := Imf. Then the following are equivalent:<br />
(i) f is complex differentiable at z0;<br />
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