Sideway BICK BlogSideway BICK BLOG from Sideway

A Sideway to Sideway Home
Home
Output.to

Link:http://output.to/sideway/default.asp?qno=190300020

Complex Analysis Complex Number Complex Derivative

source/reference:
https://www.youtube.com/channel/UCaTLkDn9_1Wy5TRYfVULYUw/playlists

Complex Derivative

Derivative of a Function

Let 𝑓:(π‘Ž,𝑏)→ℝ be a real-valued function of a real variable, and let π‘₯0∈(π‘Ž,𝑏). The function 𝑓 is differentiable at π‘₯0 if limπ‘₯β†’π‘₯0 𝑓(π‘₯)βˆ’π‘“(π‘₯0)π‘₯βˆ’π‘₯0 exist. If so, we call this limit the derivative of 𝑓 at π‘₯0 and dente it by 𝑓'(π‘₯0).

𝑓(π‘₯)βˆ’π‘“(π‘₯0)π‘₯βˆ’π‘₯0 is the slope of the secant line through the points (π‘₯0, 𝑓(π‘₯0)) and (π‘₯, 𝑓(π‘₯)). The slope of the secant line changes as π‘₯ approaches π‘₯0. In the limit, the slopes approach the slope of the tangent line to the graph of 𝑓 at π‘₯0.

However, the derivative does not always exist. For exampe, the graph of 𝑓 does not have a tangent line at π‘₯0.

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)βˆ’π‘“(𝑧0)π‘§βˆ’π‘§0 exist.

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

Example: 𝑓(𝑧)=𝖼 (a constant function, π–Όβˆˆβ„‚).

Let 𝑧0∈β„‚ be arbitrary. Then 𝑓(𝑧)βˆ’π‘“(𝑧0)π‘§βˆ’π‘§0= π–Όβˆ’π–Όπ‘§βˆ’π‘§0=0β†’0 as 𝑧→𝑧0

Thus 𝑓'(𝑧)=0 for all π‘§βˆˆβ„‚.

Other Forms of the Difference Quotient

Instead of using 𝑓(𝑧)βˆ’π‘“(𝑧0)π‘§βˆ’π‘§0

Also often write as 𝑧=𝑧0+𝗁 (where π—βˆˆβ„‚), and the difference quotient becomes

𝑓(𝑧0+β„Ž)βˆ’π‘“(𝑧0)β„Ž or simply 𝑓(𝑧+β„Ž)βˆ’π‘“(𝑧)β„Ž

where take the limit as β„Žβ†’0.

Further examples: 𝑓(𝑧)=𝑧. Then

𝑓(𝑧0+β„Ž)βˆ’π‘“(𝑧0)β„Ž= (𝑧0+β„Ž)βˆ’π‘§0β„Ž= β„Žβ„Ž=1β†’1 as β„Žβ†’0

So 𝑓′(𝑧)=1 for all π‘§βˆˆβ„‚.

More examples: 𝑓(𝑧)=𝑧2. Then

𝑓(𝑧0+β„Ž)βˆ’π‘“(𝑧0)β„Ž= (𝑧0+β„Ž)2βˆ’π‘§02β„Ž= 2𝑧0β„Ž+β„Ž2β„Ž=2𝑧0+β„Žβ†’2𝑧0 as β„Žβ†’0

Thus 𝑓′(𝑧)=2𝑧 for all π‘§βˆˆβ„‚.

Another examples: 𝑓(𝑧)=𝑧𝑛. Then

𝑓(𝑧0+β„Ž)βˆ’π‘“(𝑧0)β„Ž= (𝑧0+β„Ž)π‘›βˆ’π‘§0π‘›β„Ž=(𝑧0𝑛+π‘›β„Žπ‘§0𝑛-1+𝑛(𝑛-1) 2β„Ž2𝑧0𝑛-2+β‹―+β„Žπ‘›)βˆ’π‘§0π‘›β„Ž
=𝑛𝑧0𝑛-1+𝑛(𝑛-1) 2β„Žπ‘§0𝑛-2+β‹―+β„Žπ‘›-1=𝑛𝑧0𝑛-1+β„Ž(𝑛(𝑛-1) 2𝑧0𝑛-2+β‹―+β„Žπ‘›-2)→𝑛𝑧0𝑛-1 as β„Žβ†’0

Thus 𝑓′(𝑧)=𝑛𝑧𝑛-1 for all π‘§βˆˆβ„‚.

Differentiation Rules

By theorem. Suppose 𝑓 and 𝑔 are differentiable at 𝑧, and β„Ž is differentiable at 𝑓(𝑧). Let π‘βˆˆβ„‚. Then

  • (𝑐𝑓)β€²(𝑧)=𝑐𝑓′(𝑧)
  • (𝑓+𝑔)β€²(𝑧)=𝑓′(𝑧)+𝑔′(𝑧)
  • (𝑓*𝑔)β€²(𝑧)=𝑓′(𝑧)𝑔(𝑧)+𝑓(𝑧)𝑔′(𝑧) Product Rule
  • (𝑓𝑔)β€²(𝑧)= 𝑔(𝑧)𝑓′(𝑧)βˆ’π‘“(𝑧)𝑔′(𝑧)(𝑔(𝑧))2, for 𝑔(𝑧)β‰ 0 Quotient Rule
  • (β„Žβˆ˜π‘“)β€²(𝑧)=β„Žβ€²(𝑓(𝑧))𝑓′(𝑧) Chain Rule

Differentiability of a Function

Differentiable example

  • 𝑓(𝑧)=5𝑧3+s𝑧2-𝑧+7 then 𝑓′(𝑧)=5β‹…3𝑧2+2β‹…2π‘§βˆ’1=15𝑧2+4π‘§βˆ’1
  • 𝑓(𝑧)=1𝑧 then 𝑓′(𝑧)=𝑧⋅0βˆ’1β‹…1 𝑧2=βˆ’1𝑧2
  • 𝑓(𝑧)=(𝑧2βˆ’1)𝑛 then 𝑓′(𝑧)=𝑛(𝑧2βˆ’1)π‘›βˆ’1β‹…2𝑧
  • 𝑓(𝑧)=(𝑧2βˆ’1)(3𝑧+4) then 𝑓′(𝑧)=(2𝑧)(3𝑧+4)+(𝑧2βˆ’1)β‹…3
  • 𝑓(𝑧)=𝑧𝑧2+1 then 𝑓′(𝑧)= (𝑧2+1)βˆ’π‘§β‹…2𝑧(𝑧2+1)2= 1βˆ’π‘§2(1+𝑧2)2

Non-differentiable example

  • Let 𝑓(𝑧)=Re (𝑧). Write 𝑧=π‘₯+𝑖𝑦 and β„Ž=β„Žπ‘₯+π‘–β„Žπ‘¦. Then

    𝑓(𝑧+β„Ž)βˆ’π‘“(𝑧)β„Ž= (π‘₯+β„Žπ‘₯)βˆ’π‘₯β„Ž= β„Žπ‘₯β„Ž= Re β„Žβ„Ž

    Does 𝑓(𝑧) have a limit as β„Žβ†’0?

    • β„Žβ†’0 along real axis: Then β„Ž=β„Žπ‘₯+𝑖⋅0 , so Re β„Ž=β„Ž, and thus the quotient evaluates to 1, and the limit equals 1.
    • β„Žβ†’0 along imaginary axis: Then β„Ž=0+π‘–β‹…β„Žy, so Re β„Ž=0, and thus the quotient evaluates to 0, and the limit equals 0.
    • β„Žπ‘›=𝑖𝑛𝑛, then Re β„Žπ‘› β„Žπ‘›=Re 𝑖𝑛 𝑖𝑛={1 if 𝑛 is even0 if 𝑛 is odd has no limit as nβ†’βˆž.

    𝑓 is not differentiable anywhere in β„‚.

  • Let 𝑓(𝑧)=𝑧 then

    𝑓(𝑧+β„Ž)βˆ’π‘“(𝑧)β„Ž= (z+β„Ž)βˆ’zβ„Ž= β„Žβ„Ž
    • If β„Žβˆˆβ„ then β„Žβ„Ž=1β†’1 as β„Žβ†’0
    • If β„Žβˆˆπ‘–β„ then β„Žβ„Ž=βˆ’1β†’βˆ’1 as β„Žβ†’0

    Thus β„Žβ„Ž does not have a limit as β„Žβ†’0, and 𝑓 is not differentiable anywhere in β„‚.

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

Proof

lim𝑧→𝑧0 (𝑓(𝑧)βˆ’π‘“(𝑧0))=lim𝑧→𝑧0( 𝑓(𝑧)βˆ’π‘“(𝑧0)π‘§βˆ’π‘§0β‹…(π‘§βˆ’π‘§0))=𝑓′(𝑧0)β‹…0=0

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

By definition. A function 𝑓 is analytic in an open set π‘ˆβŠ‚β„‚ if 𝑓 is (complex) differentiable at each point π‘§βˆˆπ‘ˆ. A function which is analytic in all of β„‚ is called an entire function.

Examples:

  • polynomials are analytic in β„‚ (hence entire)
  • rational functions 𝑝(𝑧)π‘Ž(𝑧)are analytic wherever π‘Ž(𝑧)β‰ 0
  • 𝑓(𝑧)=𝑧 is not analytic
  • 𝑓(𝑧)=Re z is not analytic

Another examples:

Let 𝑓(𝑧)=|𝑧|2, then

𝑓(𝑧+β„Ž)βˆ’π‘“(𝑧)β„Ž= |𝑧+β„Ž|2βˆ’|𝑧|2β„Ž= (𝑧+β„Ž)(𝑧+β„Ž)βˆ’|𝑧|2β„Ž= |𝑧|2+π‘§β„Ž+β„Žπ‘§+β„Žβ„Žβˆ’|𝑧|2β„Ž= 𝑧+β„Ž+π‘§β‹…β„Žβ„Ž

Thus,

  • If 𝑧≠0 then the limit as β„Žβ†’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 β„‚
Previous Month  MAR  2019  Next Month
SMTWTFS
12
3456789
10111213141516
17181920212223
24252627282930
31

Previous Month  OCT  2016  Next Month
SMTWTFS
1
2345678
9101112131415
16171819202122
23242526272829
3031

Sideway BICK Blog

20/03

Copyright © 2000-2020 Sideway . All rights reserved Disclaimerslast modified on 26 January 2013