Differentiability and Derivatives
A real valued function of a real variable x is differentiable at x with derivative f′(x) if
f(x+∆x) = f(x) +f′(x) ∆x+ o(∆x).
We consider several types of functions:
Curves: Let z(t) = x(t) + i y(t) be a path (a complex valued function of a real variable t). Then z(t) is differentiable at t with derivative z′(t) if
z(t+∆t) = z(t) + z′(t) ∆t+ o(∆t).
In this case z′(t) = [dz/dt] = [dx/dt]+ i [dy/dt]. Moreover, the derivative can be calculated as the limit of difference quotients:
dz
dt
 =
lim
∆t → 0 
 z(t+∆t) − z(t)
∆t
 .
Complex functions of a complex variable z: Let f(z) be defined in an open set. Then f is differentiable at z with derivative f′(z) if
f(z+∆z) = f(x) +f′(z) ∆z+ o(∆z).
The derivative can be calculated as the limit of difference quotients:
f′(z) =
lim
∆z → 0 
 f(z+∆z) − f(z)
∆z
 .
While the formal definitions are the same, the requirements are quite different. If the variable, e.g., x, is real, there are only two ways that ∆x → 0 - from the right (∆x > 0) or from the left (∆x > 0). In the complex variable case, |∆z|→ 0, but ∆z is allowed to have random directions as ∆z → 0. The existence of a limit limcomplex z → · is stronger than the concept limreal x → ·.


If f is a differentiable function of the complex variable z in an open set or region, f(z) is also called an analytic or holomorphic function1.


Real [Complex] Functions of Two Variables (x,y): There is another concept of differentiability of functions of two (or more) variables (x,y). For simplicity write P = (x,y) and ∆P = (∆x,∆y). Then a real (or complex) valued function G is differentiable at P if
G(P + ∆P) = G(P) + linear function of ∆P + o(∆P).
All linear functions of ∆P = (∆x,∆y) are of the form a ·∆x+ b ·∆y. The numbers a and b can be calculated as the partial derivatives of G:
a = ∂G
∂x
 =
lim
∆x → 0 
 G(x+∆x,y)−G(x,y)
∆x
 ,

b = ∂G
∂y
 =
lim
∆y → 0 
 G(x,y+∆y)−G(x,y)
∆y
 .
The derivative of G at P is then the (complex) pair < [(∂G)/(∂ x)],[(∂G)/(∂y)] > so that
G(P + ∆P) = G(P) + ∂G
∂ x
 ·∆x+ ∂G
∂y
 ·∆y+ o(|∆P|).
The [complex] pair < [(∂G)/(∂ x)],[(∂G)/(∂y)] > is called the [ complex ] gradient of G and is written grad G or ∇G.
The formal expression [(∂G)/(∂ x)]  dx + [(∂G)/(∂y)]  dy is also called the differential dG of the function G(x,y). We write
dG = ∂G
∂ x
  dx + ∂G
∂y
  dy.
Back to Analytic (Holomorphic2) Functions: That a differentiable function G of two real variables P=(x,y) is a differentiable function of the complex variable z or analytic (holomorphic ) reduces to the statement:
If P=(x,y), z=x+iy, there is a complex number G′(z) such that
G′(z) ·(∆x+ i ∆y) = ∂G
∂ x
 ·∆x+ ∂G
∂y
 ·∆y.
The complex number G′(z) can be calculated several ways:
•
Let ∆y =0 and ∆x ≠ 0:
G′(z) = ∂G
∂ x
•
Let ∆y ≠ 0 and ∆x = 0:
iG′(z) = ∂G
∂ y
 , or G′(z) = −i ∂G
∂ y
•

G′(z) = ∂G
∂z
    def
≡
 
    1
2
 ⎛
⎝ 		∂G
∂ x
 −i ∂G
∂ y
⎞
⎠
The formal expression [(∂G)/(∂ x)]  dx + [(∂G)/(∂y)]  dy is also called the differential dG of the function G(x,y). If G is a differentiable function of the complex variable z, we can write formally:
dG = ∂G
∂ x
  dx + ∂G
∂y
  dy = dG
dz
  dz = dG
dz
  dz.
The precise meaning of the statement dG = [dG/dz] dz is that
G(z + ∆z) − G(z) = dG
dz
 ·∆z+ o(∆z).
Remarks on Analyticity and Partial Differential Equations (PDE)
If G(x,y) is real differentiable,
dG
= ∂G
∂x
  dx + ∂G
∂y
  dy,
dx
= 1
2
 ⎛
⎝ 		dz + d
-
z
 
⎞
⎠ 		,
dy
= 1
2i
 ⎛
⎝ 		dz − d
-
z
 
⎞
⎠ 		,
so that
dG
= 1
2
 ⎛
⎝ 		∂G
∂x
 − i ∂G
∂y
⎞
⎠ 		  dz + 1
2
 ⎛
⎝ 		∂G
∂x
 + i ∂G
∂y
⎞
⎠ 		  d
-
z
 
= ∂G
∂z
  dz + ∂G
∂
-
z
 
  d
-
z
 
 .
Thus the analytic functions are the real differentiable functions G(x,y) which satisfy the partial differential equation
∂G
∂
-
z
 
 = 1
2
 ⎛
⎝ 		∂G
∂x
 + i ∂G
∂y
⎞
⎠ 		=0.

Footnotes:

1See Knopp, Elements of the Theory of Functions, p. 101, and Knopp, Theory of Functions, Part I, p. 27. Thus if f(z) is analytic at a point z0, f(z) is actually analytic in a neighborhood of z0.
2 The words holomorphic and analytic are used interchangably.

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On 16 Nov 2013, 20:45.