Unformatted text preview: =∂y
∂x If f: DÆC is analytic, then the partial derivatives
I
∂u ∂v
=
∂x ∂y and Conversely, if u(x, y) and v(x, y) are have continuous firstorder partial derivatives in D
and satisfy the CauchyRiemann conditions on D, then f is analytic in D with
∂u
∂v ∂u
∂u
∂v
∂u ∂v
∂v
f'(z) =
+i
=
i
=
i
=
+i
∂x
∂x ∂x
∂y
∂y
∂y ∂y
∂x
Note that the second equation just above says that
f'(z) is the complex conjugate of the gradient of u(x, y)
Proof Suppose f: DÆ C is analytic. Then look at the real and imaginary parts of f'(z)
I
using ∆z = ∆x, and ∆z = i∆y. We find:
∂u
∂v
∆z = ∆x:
f'(z) =
+i
∂x
∂x
∂v
∂u
∆z = i∆y
f'(z) =
i
∂y
∂y
Equating coefficients gives us the result.
Proving the converse is beyond the scope of this course. (Basically, one proves
that the above formula for f'(z) works as a derivative.)
Examples
2
2
Show that f(z) = x  y i is nowhere analytic.
Now let us fiddle with the CR equations. Start with
∂u ∂v
∂u
∂v
=
and
=∂x ∂y
∂y
∂x
and take ∂/∂x of both sides of the first, and ∂/∂y of the second:
2
2
2
2
∂u
∂v
∂u
∂v
and
2 = ∂x∂y
2 =  ∂x∂y
∂x
∂y
Combining these gives
2 ∂u 2 ∂u 2+
2=1
∂x
∂y
u is harmonic Similarly, we see that v is harmo...
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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.
 Fall '03
 StefanWaner
 Math, Algebra, Geometry, Complex Numbers

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