Cauchy--Riemann equations

Given a holomorphic function $f : Ω \to C$ with $f (z) = (u (z), v (z))$ the following relations hold:

\begin{matrix} u_{x} (x_{0}, y_{0}) = v_{y} (x_{0}, y_{0}), \\ u_{y} (x_{0}, y_{0}) = - v_{x} (x_{0}, y_{0}) . \end{matrix}

Interpretation: as a function of two variables $f : R^{2} \to R^{2}$ we have the linear map $d f_{(x_{0}, y_{0})} : R^{2} \to R^{2}$ given by the Jacobian matrix:

J_{f} (x_{0}, y_{0}) = (\begin{matrix} u_{x} (x_{0}, y_{0}) & u_{y} (x_{0}, y_{0}) \\ v_{x} (x_{0}, y_{0}) & v_{y} (x_{0}, y_{0}) \end{matrix}) $ $ A p p l y i n g t h e C a u c h y - R i e m a n n e q u a t i o n s, w e c a n r e w r i t e t h i s m a t r i x a s :

J_f(x_0, y_0) = \begin{pmatrix}
u_x(x_0, y_0) & -v_x(x_0, y_0) \
v_x(x_0, y_0) & u_x(x_0, y_0)
\end

T h i s f o r m i s p r e c i s e l y t h e m a t r i x r e p r e s e n t a t i o n o f m u l t i p l i c a t i o n b y a c o m p l e x n u m b e r . I f w e c o n s i d e r a c o m p l e x n u m b e r $ a + b i $, i t s a c t i o n o n a n o t h e r c o m p l e x n u m b e r $ x + y i $ (r e p r e s e n t e d a s a v e c t o r $ (x, y) $) c a n b e w r i t t e n a s :

(a+bi)(x+yi) = (ax-by) + (ay+bx)i

I n m a t r i x f o r m, t h i s c o r r e s p o n d s t o :

\begin{pmatrix}
a & -b \
b & a
\end{pmatrix}
\begin{pmatrix}
x \
y
\end{pmatrix} = \begin{pmatrix}
ax-by \
bx+ay
\end

C o m p a r i n g t h i s w i t h t h e J a c o b i a n m a t r i x, w e s e e t h a t $ u_{x} (x_{0}, y_{0}) $ p l a y s t h e r o l e o f $ a $ a n d $ v_{x} (x_{0}, y_{0}) $ p l a y s t h e r o l e o f $ b $ . T h e r e f o r e, t h e d i f f e r e n t i a l $ d f_{(x_{0}, y_{0})} $ a t a p o i n t $ (x_{0}, y_{0}) $ a c t s a s m u l t i p l i c a t i o n b y t h e c o m p l e x n u m b e r $ u_{x} (x_{0}, y_{0}) + i v_{x} (x_{0}, y_{0}) $ . T h i s c o m p l e x n u m b e r i s p r e c i s e l y $ f^{'} (z_{0}) $ w h e r e $ z_{0} = x_{0} + i y_{0} $ . I n e s s e n c e, t h e h o l o m o r p h i c i t y c o n d i t i o n e n s u r e s t h a t t h e l o c a l l i n e a r a p p r o x i m a t i o n o f t h e f u n c t i o n (i t s d i f f e r e n t i a l) b e h a v e s e x a c t l y l i k e m u l t i p l i c a t i o n b y a s i n g l e c o m p l e x n u m b e r, r a t h e r t h a n a m o r e g e n e r a l l i n e a r t r a n s f o r m a t i o n i n $ R^{2} $ : i f $ f^{'} (z_{0}) \neq 0 $, t h e l o c a l l i n e a r a p p r o x i m a t i o n i s a d i l a t i o n f o l l o w e d b y a r o t a t i o n (* * a m p l i t w i s t * *, a c c o r d i n g t o [[B i b l i o g r a p h y / @ N e e d h a m 1997 V i s u a l ∥ @ N e e d h a m 1997 V i s u a l]]) .