Sideway from Sideway
Complex Analysis

Draft for Information Only


Complex Derivative
 Derivative of a Function
 The Complex Derivative
  Other Forms of the Difference Quotient
 Differentiation Rules
 Differentiability of a Function


Complex Derivative

Derivative of a Function

Let 𝑓:(𝑎,𝑏)→ℝ be a real-valued function of a real variable, and let 𝑥0∈(𝑎,𝑏). The function 𝑓 is differentiable at 𝑥0 if lim𝑥→𝑥0 𝑓(𝑥)−𝑓(𝑥0)𝑥−𝑥0 exist. If so, we call this limit the derivative of 𝑓 at 𝑥0 and dente it by 𝑓'(𝑥0).

𝑓(𝑥)−𝑓(𝑥0)𝑥−𝑥0 is the slope of the secant line through the points (𝑥0, 𝑓(𝑥0)) and (𝑥, 𝑓(𝑥)). The slope of the secant line changes as 𝑥 approaches 𝑥0. In the limit, the slopes approach the slope of the tangent line to the graph of 𝑓 at 𝑥0.

However, the derivative does not always exist. For exampe, the graph of 𝑓 does not have a tangent line at 𝑥0.

The Complex Derivative

By definition. A complex-valued function 𝑓 of a complex variable is (complex) differentiable at 𝑧0∈domain(𝑓) if lim𝑧→𝑧0 𝑓(𝑧)−𝑓(𝑧0)𝑧−𝑧0 exist.

If this limit exist, it is denoted 𝑓′(𝑧0) or 𝖽𝑓𝖽𝑧(𝑧0), or 𝖽𝖽𝑧𝑓(𝑧)

Example: 𝑓(𝑧)=𝖼 (a constant function, 𝖼∈ℂ).

Let 𝑧0∈ℂ be arbitrary. Then 𝑓(𝑧)−𝑓(𝑧0)𝑧−𝑧0= 𝖼−𝖼𝑧−𝑧0=0→0 as 𝑧→𝑧0

Thus 𝑓'(𝑧)=0 for all 𝑧∈ℂ.

Other Forms of the Difference Quotient

Instead of using 𝑓(𝑧)−𝑓(𝑧0)𝑧−𝑧0

Also often write as 𝑧=𝑧0+𝗁 (where 𝗁∈ℂ), and the difference quotient becomes

𝑓(𝑧0+ℎ)−𝑓(𝑧0) or simply 𝑓(𝑧+ℎ)−𝑓(𝑧)

where take the limit as ℎ→0.

Further examples: 𝑓(𝑧)=𝑧. Then

𝑓(𝑧0+ℎ)−𝑓(𝑧0)= (𝑧0+ℎ)−𝑧0= =1→1 as ℎ→0

So 𝑓′(𝑧)=1 for all 𝑧∈ℂ.

More examples: 𝑓(𝑧)=𝑧2. Then

𝑓(𝑧0+ℎ)−𝑓(𝑧0)= (𝑧0+ℎ)2−𝑧02= 2𝑧0ℎ+ℎ2=2𝑧0+ℎ→2𝑧0 as ℎ→0

Thus 𝑓′(𝑧)=2𝑧 for all 𝑧∈ℂ.

Another examples: 𝑓(𝑧)=𝑧𝑛. Then

𝑓(𝑧0+ℎ)−𝑓(𝑧0)= (𝑧0+ℎ)𝑛−𝑧0𝑛=(𝑧0𝑛+𝑛ℎ𝑧0𝑛-1+𝑛(𝑛-1) 22𝑧0𝑛-2+⋯+ℎ𝑛)−𝑧0𝑛
=𝑛𝑧0𝑛-1+𝑛(𝑛-1) 2ℎ𝑧0𝑛-2+⋯+ℎ𝑛-1=𝑛𝑧0𝑛-1+ℎ(𝑛(𝑛-1) 2𝑧0𝑛-2+⋯+ℎ𝑛-2)→𝑛𝑧0𝑛-1 as ℎ→0

Thus 𝑓′(𝑧)=𝑛𝑧𝑛-1 for all 𝑧∈ℂ.

Differentiation Rules

By theorem. Suppose 𝑓 and 𝑔 are differentiable at 𝑧, and ℎ is differentiable at 𝑓(𝑧). Let 𝑐∈ℂ. Then

  • (𝑐𝑓)′(𝑧)=𝑐𝑓′(𝑧)
  • (𝑓+𝑔)′(𝑧)=𝑓′(𝑧)+𝑔′(𝑧)
  • (𝑓*𝑔)′(𝑧)=𝑓′(𝑧)𝑔(𝑧)+𝑓(𝑧)𝑔′(𝑧) Product Rule
  • (𝑓𝑔)′(𝑧)= 𝑔(𝑧)𝑓′(𝑧)−𝑓(𝑧)𝑔′(𝑧)(𝑔(𝑧))2, for 𝑔(𝑧)≠0 Quotient Rule
  • (ℎ∘𝑓)′(𝑧)=ℎ′(𝑓(𝑧))𝑓′(𝑧) Chain Rule

Differentiability of a Function

Differentiable example

  • 𝑓(𝑧)=5𝑧3+s𝑧2-𝑧+7 then 𝑓′(𝑧)=5⋅3𝑧2+2⋅2𝑧−1=15𝑧2+4𝑧−1
  • 𝑓(𝑧)=1𝑧 then 𝑓′(𝑧)=𝑧⋅0−1⋅1 𝑧2=−1𝑧2
  • 𝑓(𝑧)=(𝑧2−1)𝑛 then 𝑓′(𝑧)=𝑛(𝑧2−1)𝑛−1⋅2𝑧
  • 𝑓(𝑧)=(𝑧2−1)(3𝑧+4) then 𝑓′(𝑧)=(2𝑧)(3𝑧+4)+(𝑧2−1)⋅3
  • 𝑓(𝑧)=𝑧𝑧2+1 then 𝑓′(𝑧)= (𝑧2+1)−𝑧⋅2𝑧(𝑧2+1)2= 1−𝑧2(1+𝑧2)2

Non-differentiable example

  • Let 𝑓(𝑧)=Re (𝑧). Write 𝑧=𝑥+𝑖𝑦 and ℎ=ℎ𝑥+𝑖ℎ𝑦. Then

    𝑓(𝑧+ℎ)−𝑓(𝑧)= (𝑥+ℎ𝑥)−𝑥= 𝑥= Re

    Does 𝑓(𝑧) have a limit as ℎ→0?

    • ℎ→0 along real axis: Then ℎ=ℎ𝑥+𝑖⋅0 , so Re ℎ=ℎ, and thus the quotient evaluates to 1, and the limit equals 1.
    • ℎ→0 along imaginary axis: Then ℎ=0+𝑖⋅ℎy, so Re ℎ=0, and thus the quotient evaluates to 0, and the limit equals 0.
    • 𝑛=𝑖𝑛𝑛, then Re𝑛 𝑛=Re 𝑖𝑛 𝑖𝑛={1 if 𝑛 is even0 if 𝑛 is odd has no limit as n→∞.

    𝑓 is not differentiable anywhere in ℂ.

  • Let 𝑓(𝑧)=𝑧 then

    𝑓(𝑧+ℎ)−𝑓(𝑧)= (z+)−z=
    • If ℎ∈ℝ then =1→1 as ℎ→0
    • If ℎ∈𝑖ℝ then =−1→−1 as ℎ→0

    Thus does not have a limit as ℎ→0, and 𝑓 is not differentiable anywhere in ℂ.

By Fact. If 𝑓 is differentiable at z0 then 𝑓 is continuos at 𝑧0.


lim𝑧→𝑧0 (𝑓(𝑧)−𝑓(𝑧0))=lim𝑧→𝑧0( 𝑓(𝑧)−𝑓(𝑧0)𝑧−𝑧0⋅(𝑧−𝑧0))=𝑓′(𝑧0)⋅0=0

Note however that a function can be continuous without being differentiable.

By definition. A function 𝑓 is analytic in an open set 𝑈⊂ℂ if 𝑓 is (complex) differentiable at each point 𝑧∈𝑈. A function which is analytic in all of ℂ is called an entire function.


  • polynomials are analytic in ℂ (hence entire)
  • rational functions 𝑝(𝑧)𝑎(𝑧)are analytic wherever 𝑎(𝑧)≠0
  • 𝑓(𝑧)=𝑧 is not analytic
  • 𝑓(𝑧)=Re z is not analytic

Another examples:

Let 𝑓(𝑧)=|𝑧|2, then

𝑓(𝑧+ℎ)−𝑓(𝑧)= |𝑧+ℎ|2−|𝑧|2= (𝑧+ℎ)(𝑧+)−|𝑧|2= |𝑧|2+𝑧+ℎ𝑧+ℎ−|𝑧|2= 𝑧++𝑧⋅


  • If 𝑧≠0 then the limit as ℎ→0 does not exist.
  • If 𝑧=0 then the limit equals 0, thus 𝑓 is differentiable at 0 with 𝑓′(𝑧)=0.
  • 𝑓 is not analytic anywhere
  • Note: 𝑓 is continuous in ℂ


ID: 190300020 Last Updated: 2019/3/20 Revision:


Home (5)



HBR (3)



Hobbies (7)


Chinese (1097)

English (336)

Reference (66)


Hardware (149)


Application (187)

Digitization (24)

Numeric (19)


Web (644)new



Regular Expression (SC)


Knowledge Base

Common Color (SC)

Html Entity (Unicode) (SC)

Html 401 Special (SC)

OS (389)new

MS Windows

Windows10 (SC)

.NET Framework (SC)

DeskTop (7)



Formulas (8)

Number Theory (206)

Algebra (20)

Trigonometry (18)

Geometry (18)

Calculus (67)

Complex Analysis (21)


Tables (8)


Mechanics (1)

Rigid Bodies

Statics (92)

Dynamics (37)

Fluid (5)

Fluid Kinematics (5)


Process Control (1)

Acoustics (19)

FiniteElement (2)

Biology (1)

Geography (1)

Latest Updated Links

Copyright © 2000-2019 Sideway . All rights reserved Disclaimers last modified on 10 Feb 2019