Differentiation Examples and Rules
•
The power functions zn, n = 0, ±1, ±2, …, are analytic and
dzn
dz
 = n zn−1.
•
The conjugate function, g(z) = ―z, is not differentiable for any z.
Note that

lim
z → 0 
-
z
z
does not exist.
•
Cauchy - Riemann Equations: If G(x,y) = u(x,y) + i v(x,y) is a complex valued function of two variable with continuous partial derivatives, then G(z) = G(x + i y) is analytic if and only if
∂G
∂ x
 = −i ∂G
∂ y
 = G′(z).
In terms of real and imaginary parts,
∂u
∂x
= ∂v
∂ y
 ,
∂u
∂y
= − ∂v
∂x
 .
•
Verify that the following functions satisfy the Cauchy - Riemann Equations:
•

exp(x + i y) ≡ ex(cos(y) + i sin(y)).
•

Ln(x + i y) ≡ ln|x+iy| + i arctan⎛
⎝ 		y
x
⎞
⎠ 		, x > 0.
•
Verify that
•

d exp(z)
dz
 = exp
(z).
•

d Ln
(z)
dz
 = 1
z
 , ℜz > 0.
•
The Chain Rule. Let g(z) be analytic at z, and let f(w) be analytic at w = g(z). Then h(z) = f(g(z)) is analytic at z and
d
dz
 f(g(z)) = f′(g(z))·g′(z).
Proof: We are assuming that g(z+∆z) = g(z) +g′(z) ·∆z+ o(∆z) and f(g(z)+∆g(z)) = f(g(z)) +f′(g(z)) ·∆g(z) + o(∆ g(z)). Since ∆g(z) = g′(z) ∆z+ o(∆z), and o(∆ g(z)) = o(O(∆z)), we have f(g(z + ∆z)) = f(g(z)) +f′(g(z)) ·g′(z) ·∆z+ o(∆z).


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On 16 Nov 2013, 20:47.