SlideShare a Scribd company logo
1 of 12
Download to read offline
POWER SERIES
(TOPIC NO. 14)
NAME: JAIMIN AJITKUMAR KEMKAR
EN. NO.: 160123119014
BRANCH: MECHANICAL
SEM. & CLASS: 4D
SUBJECT: COMPLEX VARIABLES AND NUMERIC METHODS (2141905)
GUIDED BY: PROF. KEYURI M. SHAH
CONTENTS
• SINGULAR POINT AND CLASSIFICATIONS
• RESIDUE THEOREM
• ROUCHE’S THEOREM (WITHOUT PROOF)
SINGULAR POINTS
• A POINT AT WHICH THE FUNCTION 𝑓 𝑧 IS NOT ANALYTIC IS KNOWN AS
SINGULAR POINT OR SINGULARITY OF THE FUNCTION.
TYPES OF SINGULARITIES
1. ISOLATED SINGULARITY
A SINGULAR POINT 𝑧 = 𝑎 IS KNOWN AS AN ISOLATED SINGULARITY IF THERE IS NO OTHER
SINGULARITY WITHIN A SMALL CIRCLE SURROUNDING THE POINT 𝑧 = 𝑎 ; OTHERWISE IT IS CALLED A NON-
ISOLATED SINGULARITY.
EX. 𝑓 𝑧 =
𝑧2
𝑧−2
HAS ISOLATED SINGULARITY AT 𝑧 = 2.
2. REMOVABLE SINGULARITY
AN ISOLATED SINGULAR POINT 𝑧 = 𝑎 IS KNOWN AS A REMOVABLE SINGULARTITY IF lim
𝑧→𝑎
𝑓(𝑧) EXISTS
OR THERE IS NO NEGATIVE POWER TERM OF (𝑧 − 𝑎) IN LAURENT’S SERIES OF 𝑓(𝑧), I.E., 𝑏 𝑛 = 0.
EX. 𝑓 𝑧 =
sin 𝑧
𝑧
𝑧 = 0 IS A SINGULARITY OF 𝑓(𝑧).
lim
𝑧→0
sin 𝑧
𝑧
= 1
∴ 𝑓(𝑧) HAS A REMOVABLE SINGULARITY AT 𝑧 = 0.
3. ESSENTIAL SINGULARITY
A SINGULAR POINT 𝑧 = 𝑎 IS KNOWN AS AN ESSENTIAL SINGULARITY IF THE NUMBER OF NEGATIVE
POWER TERMS OF (𝑧 − 𝑎) IN LAURENT’S SERIES IS INFINITE.
EX. 𝑓 𝑧 = 𝑒
1
𝑧−2
= 1 +
1
𝑧−2
+
1
2!
1
(𝑧−2)2 + ⋯
SINCE THE NUMBER OF NEGATIVE POWER TERMS OF (𝑧 − 2) IS INFINITE, 𝑧 = 2 IS AN ESSENTIAL
SINGULARITY.
4. POLE OF ORDER m
AN ISOLATED SINGULARITY 𝑧 = 𝑎 IS KNOWN AS A POLE OF ORDER m IF IN LAURENT’S SERIES, ALL
THE NEGATIVE POWER OF (𝑧 − 𝑎) AFTER THE 𝑚 𝑡ℎ
POWER ARE ZERO, I.E., HIGHEST POWER OF
1
𝑧−𝑎
IS m.
EX. 𝑓 𝑧 =
1
(𝑧−1)(𝑧−3)2
𝑧 = 1 IS A POLE OF ORDER 1 AND 𝑧 = 3 IS A POLE ORDER 2.
NOTE: A POLE OF ORDER ONE IS ALSO KNOWN AS A SIMPLE POLE.
RESIDUE
• THE COEFFICIENT OF
1
𝑧−𝑎
IN THE LAURENT SERIES EXPANSION OF 𝑓(𝑧) IS
KNOWN AS THE RESIDUE OF 𝑓(𝑧) AT 𝑧 = 𝑎.
• THE LAURENT SERIES EXPANSION ABOUT 𝑧 = 𝑎
𝑓 𝑧 = 𝑛=0
∞
𝑎 𝑛(𝑧 − 𝑎) 𝑛
+ 𝑛=1
∞ 𝑏 𝑛
(𝑧−𝑎) 𝑛
RESIDUE AT (𝑧 = 𝑎)= COEFFICIENT OF
1
𝑧−𝑎
= 𝑏1
=
1
2𝜋𝑖 𝑐
𝑓(𝑧)
(𝑧−𝑎)−1+1 𝑑𝑧 𝑏 𝑛 =
1
2𝜋𝑖
𝑓(𝑧)
(𝑧−𝑎)−𝑛+1 𝑑𝑧
=
1
2𝜋𝑖 𝑐
𝑓 𝑧 𝑑𝑧
EVOLUATION OF RESIDUES
1. RESIDUE AT A SIMPLE POLE
i. IF 𝑓(𝑧) HAS A SIMPLE POLE AT 𝑧 = 𝑎 THEN
𝑅𝑒𝑠 𝑓 𝑧 ; 𝑧 = 𝑎 = lim
𝑧→𝑎
𝑧 − 𝑎 𝑓(𝑧)
ii. IF 𝑓 𝑧 IS OF THE FORM 𝑓 𝑧 =
𝑔(𝑧)
ℎ(𝑧)
, WHERE 𝑔(𝑧) AND ℎ(𝑧) ARE ANALYTIC AT
𝑧 = 𝑎.
IF ℎ 𝑎 = 0 BUT ℎ′(𝑎) ≠ 0 THEN 𝑧 = 𝑎 IS A SIMPLE POLE.
IF ℎ 𝑎 = 0 NUT 𝑔(𝑎) ≠ 0 THEN
𝑅𝑒𝑠 𝑓 𝑧 ; 𝑧 = 𝑎 =
1
(𝑚−1)!
lim
𝑧→𝑎
𝑔(𝑧)
ℎ′(𝑧)
2. RESIDUE AT A POLE OF ORDER m
IF 𝑓(𝑧)HAS A POLE OF ORDER m AT 𝑧 = 𝑎 THEN
𝑅𝑒𝑠 𝑓 𝑧 ; 𝑧 = 𝑎 =
1
(𝑚−1)!
lim
𝑧→𝑎
𝑑 𝑚−1
𝑑𝑧 𝑚−1 𝑧 − 𝑎 𝑚
𝑓(𝑧)
ROUCHE’S THEOREM
• LET 𝑓(𝑧) AND 𝑔(𝑧) BE ANALYTIC IN A DOMAIN D AND HAVE FINITE NUMBER
OF ZEROS IN D.
• SUPPOSE C IS A SIMPLE CLOSED CONTOUR IN D SUCH THAT NO ZEROS OF
𝑓(𝑧) AN 𝑔(𝑧) LIE ON C AND 𝑔(𝑧) < 𝑓(𝑧) ON C.
• THEN THE TOTAL NUMBER OF ZEROS OF 𝑓 𝑧 + 𝑔(𝑧) IS EQUAL TO THE TOTAL
NUMBER OF ZEROS OF 𝑓(𝑧) INSIDE C.
• I.E., 𝑁𝑓+𝑔 = 𝑁𝑓
EX. USE ROUCHE’S THEOREM TO DETERMINE THE NUMBER OF ZEROS OF THE
POLYNOMIAL 𝒛 𝟔 + 𝟓𝒛 𝟒 + 𝒛 𝟑 + 𝟐𝒛 INSIDE THE CIRCLE 𝒛 = 𝟏
LET 𝑓 𝑧 = 𝒛 𝟔
+ 𝟓𝒛 𝟒
+ 𝒛 𝟑
+ 𝟐𝒛
LET 𝑓 𝑧 = −5𝑧4 , 𝑔 𝑧 = 𝑧6 + 𝑧3 + 2𝑧 , 𝐶: 𝑧 = 1
ON CIRCLE 𝐶: 𝑧 = 1
𝑓 𝑧 = −5𝑧4
= 5
And 𝑔(𝑧) = 𝑧6
+ 𝑧3
− 2𝑧
≤ 𝑧6
+ 𝑧3
+ 2 𝑧
= 4
∴ 𝑔(𝑧) ≤ 4
∴ 𝑓(𝑧) = 5 > 4 ≥ 𝑔(𝑧)
∴ 𝑓(𝑧) > 𝑔(𝑧)
SINCE 𝑓(𝑧) AND 𝑔(𝑧) ARE ANALYTIC WITHIN AND ON C, ALSO 𝑓(𝑧) AND 𝑓 𝑧 + 𝑔(𝑧) ARE
NONZERO ON C, BY ROUCHE’S THEOREM,
𝑁𝑓+𝑔 = 𝑁𝑓
𝑓 𝑧 = −5𝑧4 HAS A ZERO OF ORDER 4 INCIDE C.
𝑁𝑓 = 4
∴ 𝑁𝑓+𝑔 = 4
HENCE, 𝑓 𝑧 HAS FOUR ZEROS INSIDE C.
REFERANCES
• BOOK: Mc Graw Hill (CVNM)
Power series

More Related Content

What's hot

Complex analysis
Complex analysisComplex analysis
Complex analysissujathavvv
 
SOP POS, Minterm and Maxterm
SOP POS, Minterm and MaxtermSOP POS, Minterm and Maxterm
SOP POS, Minterm and MaxtermSelf-employed
 
Inner product spaces
Inner product spacesInner product spaces
Inner product spacesEasyStudy3
 
Vector Spaces,subspaces,Span,Basis
Vector Spaces,subspaces,Span,BasisVector Spaces,subspaces,Span,Basis
Vector Spaces,subspaces,Span,BasisRavi Gelani
 
SERIES SOLUTION OF ORDINARY DIFFERENTIALL EQUATION
SERIES SOLUTION OF ORDINARY DIFFERENTIALL EQUATIONSERIES SOLUTION OF ORDINARY DIFFERENTIALL EQUATION
SERIES SOLUTION OF ORDINARY DIFFERENTIALL EQUATIONKavin Raval
 
signal & system inverse z-transform
signal & system inverse z-transformsignal & system inverse z-transform
signal & system inverse z-transformmihir jain
 
Sequences and Series (Mathematics)
Sequences and Series (Mathematics) Sequences and Series (Mathematics)
Sequences and Series (Mathematics) Dhrumil Maniar
 
Series solution to ordinary differential equations
Series solution to ordinary differential equations Series solution to ordinary differential equations
Series solution to ordinary differential equations University of Windsor
 
Complex Analysis - Differentiability and Analyticity (Team 2) - University of...
Complex Analysis - Differentiability and Analyticity (Team 2) - University of...Complex Analysis - Differentiability and Analyticity (Team 2) - University of...
Complex Analysis - Differentiability and Analyticity (Team 2) - University of...Alex Bell
 
Discrete Fourier Transform
Discrete Fourier TransformDiscrete Fourier Transform
Discrete Fourier TransformAbhishek Choksi
 
Complex analysis
Complex analysisComplex analysis
Complex analysissujathavvv
 
Z Transform And Inverse Z Transform - Signal And Systems
Z Transform And Inverse Z Transform - Signal And SystemsZ Transform And Inverse Z Transform - Signal And Systems
Z Transform And Inverse Z Transform - Signal And SystemsMr. RahüL YøGi
 
Linear dependence & independence vectors
Linear dependence & independence vectorsLinear dependence & independence vectors
Linear dependence & independence vectorsRakib Hossain
 
complex variable PPT ( SEM 2 / CH -2 / GTU)
complex variable PPT ( SEM 2 / CH -2 / GTU)complex variable PPT ( SEM 2 / CH -2 / GTU)
complex variable PPT ( SEM 2 / CH -2 / GTU)tejaspatel1997
 
Integration in the complex plane
Integration in the complex planeIntegration in the complex plane
Integration in the complex planeAmit Amola
 

What's hot (20)

Complex analysis
Complex analysisComplex analysis
Complex analysis
 
Complex analysis
Complex analysisComplex analysis
Complex analysis
 
SOP POS, Minterm and Maxterm
SOP POS, Minterm and MaxtermSOP POS, Minterm and Maxterm
SOP POS, Minterm and Maxterm
 
Inner product spaces
Inner product spacesInner product spaces
Inner product spaces
 
Vector Spaces,subspaces,Span,Basis
Vector Spaces,subspaces,Span,BasisVector Spaces,subspaces,Span,Basis
Vector Spaces,subspaces,Span,Basis
 
SERIES SOLUTION OF ORDINARY DIFFERENTIALL EQUATION
SERIES SOLUTION OF ORDINARY DIFFERENTIALL EQUATIONSERIES SOLUTION OF ORDINARY DIFFERENTIALL EQUATION
SERIES SOLUTION OF ORDINARY DIFFERENTIALL EQUATION
 
signal & system inverse z-transform
signal & system inverse z-transformsignal & system inverse z-transform
signal & system inverse z-transform
 
Sequences and Series (Mathematics)
Sequences and Series (Mathematics) Sequences and Series (Mathematics)
Sequences and Series (Mathematics)
 
Series solution to ordinary differential equations
Series solution to ordinary differential equations Series solution to ordinary differential equations
Series solution to ordinary differential equations
 
Z transform
Z transformZ transform
Z transform
 
Complex Analysis - Differentiability and Analyticity (Team 2) - University of...
Complex Analysis - Differentiability and Analyticity (Team 2) - University of...Complex Analysis - Differentiability and Analyticity (Team 2) - University of...
Complex Analysis - Differentiability and Analyticity (Team 2) - University of...
 
Discrete Fourier Transform
Discrete Fourier TransformDiscrete Fourier Transform
Discrete Fourier Transform
 
Complex analysis
Complex analysisComplex analysis
Complex analysis
 
Z Transform And Inverse Z Transform - Signal And Systems
Z Transform And Inverse Z Transform - Signal And SystemsZ Transform And Inverse Z Transform - Signal And Systems
Z Transform And Inverse Z Transform - Signal And Systems
 
Linear dependence & independence vectors
Linear dependence & independence vectorsLinear dependence & independence vectors
Linear dependence & independence vectors
 
complex variable PPT ( SEM 2 / CH -2 / GTU)
complex variable PPT ( SEM 2 / CH -2 / GTU)complex variable PPT ( SEM 2 / CH -2 / GTU)
complex variable PPT ( SEM 2 / CH -2 / GTU)
 
Integration in the complex plane
Integration in the complex planeIntegration in the complex plane
Integration in the complex plane
 
Power series
Power series Power series
Power series
 
Complex function
Complex functionComplex function
Complex function
 
Taylor series
Taylor seriesTaylor series
Taylor series
 

Similar to Power series

Mathematics of nyquist plot [autosaved] [autosaved]
Mathematics of nyquist plot [autosaved] [autosaved]Mathematics of nyquist plot [autosaved] [autosaved]
Mathematics of nyquist plot [autosaved] [autosaved]Asafak Husain
 
Contour integration and Mittag Leffler theorem
Contour integration and Mittag Leffler theoremContour integration and Mittag Leffler theorem
Contour integration and Mittag Leffler theoremAmenahGondal1
 
Lecture Notes: EEEC4340318 Instrumentation and Control Systems - System Models
Lecture Notes:  EEEC4340318 Instrumentation and Control Systems - System ModelsLecture Notes:  EEEC4340318 Instrumentation and Control Systems - System Models
Lecture Notes: EEEC4340318 Instrumentation and Control Systems - System ModelsAIMST University
 
Lecture Notes: EEEC4340318 Instrumentation and Control Systems - Root Locus ...
Lecture Notes:  EEEC4340318 Instrumentation and Control Systems - Root Locus ...Lecture Notes:  EEEC4340318 Instrumentation and Control Systems - Root Locus ...
Lecture Notes: EEEC4340318 Instrumentation and Control Systems - Root Locus ...AIMST University
 
mathemaics 10 lesson about cicles. its part
mathemaics 10 lesson about cicles. its partmathemaics 10 lesson about cicles. its part
mathemaics 10 lesson about cicles. its partReinabelleMarfilMarq
 
Further Results On The Basis Of Cauchy’s Proper Bound for the Zeros of Entire...
Further Results On The Basis Of Cauchy’s Proper Bound for the Zeros of Entire...Further Results On The Basis Of Cauchy’s Proper Bound for the Zeros of Entire...
Further Results On The Basis Of Cauchy’s Proper Bound for the Zeros of Entire...IJMER
 
Domain-Range-Intercepts-Zeros-and-Asymptotes-of-Rational-Function.pptx
Domain-Range-Intercepts-Zeros-and-Asymptotes-of-Rational-Function.pptxDomain-Range-Intercepts-Zeros-and-Asymptotes-of-Rational-Function.pptx
Domain-Range-Intercepts-Zeros-and-Asymptotes-of-Rational-Function.pptxNeomyAngelaLeono1
 
GEOMETRI ANALITIK BIDANG
GEOMETRI ANALITIK BIDANGGEOMETRI ANALITIK BIDANG
GEOMETRI ANALITIK BIDANGFebri Arianti
 
Semana 10 numeros complejos i álgebra-uni ccesa007
Semana 10   numeros complejos i álgebra-uni ccesa007Semana 10   numeros complejos i álgebra-uni ccesa007
Semana 10 numeros complejos i álgebra-uni ccesa007Demetrio Ccesa Rayme
 
Rational function 11
Rational function 11Rational function 11
Rational function 11AjayQuines
 
Business statestics formullas
Business statestics formullasBusiness statestics formullas
Business statestics formullasABHISHEK SHARMA
 
Lesson 9 transcendental functions
Lesson 9 transcendental functionsLesson 9 transcendental functions
Lesson 9 transcendental functionsLawrence De Vera
 
Digital text book
Digital text bookDigital text book
Digital text booklaluls212
 

Similar to Power series (20)

Mathematics of nyquist plot [autosaved] [autosaved]
Mathematics of nyquist plot [autosaved] [autosaved]Mathematics of nyquist plot [autosaved] [autosaved]
Mathematics of nyquist plot [autosaved] [autosaved]
 
Mod 3.pptx
Mod 3.pptxMod 3.pptx
Mod 3.pptx
 
Contour integration and Mittag Leffler theorem
Contour integration and Mittag Leffler theoremContour integration and Mittag Leffler theorem
Contour integration and Mittag Leffler theorem
 
Lecture Notes: EEEC4340318 Instrumentation and Control Systems - System Models
Lecture Notes:  EEEC4340318 Instrumentation and Control Systems - System ModelsLecture Notes:  EEEC4340318 Instrumentation and Control Systems - System Models
Lecture Notes: EEEC4340318 Instrumentation and Control Systems - System Models
 
Basic calculus (i)
Basic calculus (i)Basic calculus (i)
Basic calculus (i)
 
lec39.ppt
lec39.pptlec39.ppt
lec39.ppt
 
Helicopter rotor dynamics
Helicopter rotor dynamicsHelicopter rotor dynamics
Helicopter rotor dynamics
 
Lecture Notes: EEEC4340318 Instrumentation and Control Systems - Root Locus ...
Lecture Notes:  EEEC4340318 Instrumentation and Control Systems - Root Locus ...Lecture Notes:  EEEC4340318 Instrumentation and Control Systems - Root Locus ...
Lecture Notes: EEEC4340318 Instrumentation and Control Systems - Root Locus ...
 
lec36.ppt
lec36.pptlec36.ppt
lec36.ppt
 
mathemaics 10 lesson about cicles. its part
mathemaics 10 lesson about cicles. its partmathemaics 10 lesson about cicles. its part
mathemaics 10 lesson about cicles. its part
 
Further Results On The Basis Of Cauchy’s Proper Bound for the Zeros of Entire...
Further Results On The Basis Of Cauchy’s Proper Bound for the Zeros of Entire...Further Results On The Basis Of Cauchy’s Proper Bound for the Zeros of Entire...
Further Results On The Basis Of Cauchy’s Proper Bound for the Zeros of Entire...
 
Domain-Range-Intercepts-Zeros-and-Asymptotes-of-Rational-Function.pptx
Domain-Range-Intercepts-Zeros-and-Asymptotes-of-Rational-Function.pptxDomain-Range-Intercepts-Zeros-and-Asymptotes-of-Rational-Function.pptx
Domain-Range-Intercepts-Zeros-and-Asymptotes-of-Rational-Function.pptx
 
Chemical Bonding
Chemical BondingChemical Bonding
Chemical Bonding
 
GEOMETRI ANALITIK BIDANG
GEOMETRI ANALITIK BIDANGGEOMETRI ANALITIK BIDANG
GEOMETRI ANALITIK BIDANG
 
Semana 10 numeros complejos i álgebra-uni ccesa007
Semana 10   numeros complejos i álgebra-uni ccesa007Semana 10   numeros complejos i álgebra-uni ccesa007
Semana 10 numeros complejos i álgebra-uni ccesa007
 
lec40.ppt
lec40.pptlec40.ppt
lec40.ppt
 
Rational function 11
Rational function 11Rational function 11
Rational function 11
 
Business statestics formullas
Business statestics formullasBusiness statestics formullas
Business statestics formullas
 
Lesson 9 transcendental functions
Lesson 9 transcendental functionsLesson 9 transcendental functions
Lesson 9 transcendental functions
 
Digital text book
Digital text bookDigital text book
Digital text book
 

More from jaimin kemkar

Vibration measurements
Vibration measurementsVibration measurements
Vibration measurementsjaimin kemkar
 
Ram jet propulsive engine
Ram jet propulsive engineRam jet propulsive engine
Ram jet propulsive enginejaimin kemkar
 
Rapid transverse and feed circuit
Rapid transverse and feed circuitRapid transverse and feed circuit
Rapid transverse and feed circuitjaimin kemkar
 
Steps invoved in casting
Steps invoved in castingSteps invoved in casting
Steps invoved in castingjaimin kemkar
 
Intoduction to control system
Intoduction to control systemIntoduction to control system
Intoduction to control systemjaimin kemkar
 
Torque and power measurement
Torque and power measurementTorque and power measurement
Torque and power measurementjaimin kemkar
 
rope brake dynamometer
rope brake dynamometerrope brake dynamometer
rope brake dynamometerjaimin kemkar
 
Hemispherical pressure vessel
Hemispherical pressure vesselHemispherical pressure vessel
Hemispherical pressure vesseljaimin kemkar
 

More from jaimin kemkar (14)

Vibration measurements
Vibration measurementsVibration measurements
Vibration measurements
 
Game theory
Game theoryGame theory
Game theory
 
Ram jet propulsive engine
Ram jet propulsive engineRam jet propulsive engine
Ram jet propulsive engine
 
Rapid transverse and feed circuit
Rapid transverse and feed circuitRapid transverse and feed circuit
Rapid transverse and feed circuit
 
Steps invoved in casting
Steps invoved in castingSteps invoved in casting
Steps invoved in casting
 
Intoduction to control system
Intoduction to control systemIntoduction to control system
Intoduction to control system
 
Torque and power measurement
Torque and power measurementTorque and power measurement
Torque and power measurement
 
Thread
ThreadThread
Thread
 
rope brake dynamometer
rope brake dynamometerrope brake dynamometer
rope brake dynamometer
 
Heat transfer
Heat transferHeat transfer
Heat transfer
 
Hemispherical pressure vessel
Hemispherical pressure vesselHemispherical pressure vessel
Hemispherical pressure vessel
 
multi stage pump
multi stage pumpmulti stage pump
multi stage pump
 
INFLATION
INFLATIONINFLATION
INFLATION
 
Carbonitriding
CarbonitridingCarbonitriding
Carbonitriding
 

Recently uploaded

Detection&Tracking - Thermal imaging object detection and tracking
Detection&Tracking - Thermal imaging object detection and trackingDetection&Tracking - Thermal imaging object detection and tracking
Detection&Tracking - Thermal imaging object detection and trackinghadarpinhas1
 
Introduction to Artificial Intelligence: Intelligent Agents, State Space Sear...
Introduction to Artificial Intelligence: Intelligent Agents, State Space Sear...Introduction to Artificial Intelligence: Intelligent Agents, State Space Sear...
Introduction to Artificial Intelligence: Intelligent Agents, State Space Sear...shreenathji26
 
Livre Implementing_Six_Sigma_and_Lean_A_prac([Ron_Basu]_).pdf
Livre Implementing_Six_Sigma_and_Lean_A_prac([Ron_Basu]_).pdfLivre Implementing_Six_Sigma_and_Lean_A_prac([Ron_Basu]_).pdf
Livre Implementing_Six_Sigma_and_Lean_A_prac([Ron_Basu]_).pdfsaad175691
 
KCD Costa Rica 2024 - Nephio para parvulitos
KCD Costa Rica 2024 - Nephio para parvulitosKCD Costa Rica 2024 - Nephio para parvulitos
KCD Costa Rica 2024 - Nephio para parvulitosVictor Morales
 
Ergodomus - LOD 400 Production Drawings Exampes - Copy.pdf
Ergodomus - LOD 400 Production Drawings Exampes - Copy.pdfErgodomus - LOD 400 Production Drawings Exampes - Copy.pdf
Ergodomus - LOD 400 Production Drawings Exampes - Copy.pdfgestioneergodomus
 
TEST CASE GENERATION GENERATION BLOCK BOX APPROACH
TEST CASE GENERATION GENERATION BLOCK BOX APPROACHTEST CASE GENERATION GENERATION BLOCK BOX APPROACH
TEST CASE GENERATION GENERATION BLOCK BOX APPROACHSneha Padhiar
 
Introduction of Object Oriented Programming Language using Java. .pptx
Introduction of Object Oriented Programming Language using Java. .pptxIntroduction of Object Oriented Programming Language using Java. .pptx
Introduction of Object Oriented Programming Language using Java. .pptxPoonam60376
 
Design and Analysis of Algorithms Lecture Notes
Design and Analysis of Algorithms Lecture NotesDesign and Analysis of Algorithms Lecture Notes
Design and Analysis of Algorithms Lecture NotesSreedhar Chowdam
 
LM7_ Embedded Sql and Dynamic SQL in dbms
LM7_ Embedded Sql and Dynamic SQL in dbmsLM7_ Embedded Sql and Dynamic SQL in dbms
LM7_ Embedded Sql and Dynamic SQL in dbmsBalaKrish12
 
input buffering in lexical analysis in CD
input buffering in lexical analysis in CDinput buffering in lexical analysis in CD
input buffering in lexical analysis in CDHeadOfDepartmentComp1
 
maths mini project ( applictions of quadratic forms and SVD ).ppt
maths mini project ( applictions of quadratic forms and SVD ).pptmaths mini project ( applictions of quadratic forms and SVD ).ppt
maths mini project ( applictions of quadratic forms and SVD ).pptManavPatane
 
Analysis and Evaluation of Dal Lake Biomass for Conversion to Fuel/Green fert...
Analysis and Evaluation of Dal Lake Biomass for Conversion to Fuel/Green fert...Analysis and Evaluation of Dal Lake Biomass for Conversion to Fuel/Green fert...
Analysis and Evaluation of Dal Lake Biomass for Conversion to Fuel/Green fert...arifengg7
 
Triangulation survey (Basic Mine Surveying)_MI10412MI.pptx
Triangulation survey (Basic Mine Surveying)_MI10412MI.pptxTriangulation survey (Basic Mine Surveying)_MI10412MI.pptx
Triangulation survey (Basic Mine Surveying)_MI10412MI.pptxRomil Mishra
 
Indian Tradition, Culture & Societies.pdf
Indian Tradition, Culture & Societies.pdfIndian Tradition, Culture & Societies.pdf
Indian Tradition, Culture & Societies.pdfalokitpathak01
 
22CYT12 & Chemistry for Computer Systems_Unit-II-Corrosion & its Control Meth...
22CYT12 & Chemistry for Computer Systems_Unit-II-Corrosion & its Control Meth...22CYT12 & Chemistry for Computer Systems_Unit-II-Corrosion & its Control Meth...
22CYT12 & Chemistry for Computer Systems_Unit-II-Corrosion & its Control Meth...KrishnaveniKrishnara1
 
Madani.store - Planning - Interview Questions
Madani.store - Planning - Interview QuestionsMadani.store - Planning - Interview Questions
Madani.store - Planning - Interview QuestionsKarim Gaber
 
Overview of IS 16700:2023 (by priyansh verma)
Overview of IS 16700:2023 (by priyansh verma)Overview of IS 16700:2023 (by priyansh verma)
Overview of IS 16700:2023 (by priyansh verma)Priyansh
 
priority interrupt computer organization
priority interrupt computer organizationpriority interrupt computer organization
priority interrupt computer organizationchnrketan
 

Recently uploaded (20)

Detection&Tracking - Thermal imaging object detection and tracking
Detection&Tracking - Thermal imaging object detection and trackingDetection&Tracking - Thermal imaging object detection and tracking
Detection&Tracking - Thermal imaging object detection and tracking
 
Introduction to Artificial Intelligence: Intelligent Agents, State Space Sear...
Introduction to Artificial Intelligence: Intelligent Agents, State Space Sear...Introduction to Artificial Intelligence: Intelligent Agents, State Space Sear...
Introduction to Artificial Intelligence: Intelligent Agents, State Space Sear...
 
Livre Implementing_Six_Sigma_and_Lean_A_prac([Ron_Basu]_).pdf
Livre Implementing_Six_Sigma_and_Lean_A_prac([Ron_Basu]_).pdfLivre Implementing_Six_Sigma_and_Lean_A_prac([Ron_Basu]_).pdf
Livre Implementing_Six_Sigma_and_Lean_A_prac([Ron_Basu]_).pdf
 
KCD Costa Rica 2024 - Nephio para parvulitos
KCD Costa Rica 2024 - Nephio para parvulitosKCD Costa Rica 2024 - Nephio para parvulitos
KCD Costa Rica 2024 - Nephio para parvulitos
 
Ergodomus - LOD 400 Production Drawings Exampes - Copy.pdf
Ergodomus - LOD 400 Production Drawings Exampes - Copy.pdfErgodomus - LOD 400 Production Drawings Exampes - Copy.pdf
Ergodomus - LOD 400 Production Drawings Exampes - Copy.pdf
 
TEST CASE GENERATION GENERATION BLOCK BOX APPROACH
TEST CASE GENERATION GENERATION BLOCK BOX APPROACHTEST CASE GENERATION GENERATION BLOCK BOX APPROACH
TEST CASE GENERATION GENERATION BLOCK BOX APPROACH
 
Introduction of Object Oriented Programming Language using Java. .pptx
Introduction of Object Oriented Programming Language using Java. .pptxIntroduction of Object Oriented Programming Language using Java. .pptx
Introduction of Object Oriented Programming Language using Java. .pptx
 
Neometrix Optical Balloon Theodolite.pptx
Neometrix Optical Balloon Theodolite.pptxNeometrix Optical Balloon Theodolite.pptx
Neometrix Optical Balloon Theodolite.pptx
 
Design and Analysis of Algorithms Lecture Notes
Design and Analysis of Algorithms Lecture NotesDesign and Analysis of Algorithms Lecture Notes
Design and Analysis of Algorithms Lecture Notes
 
LM7_ Embedded Sql and Dynamic SQL in dbms
LM7_ Embedded Sql and Dynamic SQL in dbmsLM7_ Embedded Sql and Dynamic SQL in dbms
LM7_ Embedded Sql and Dynamic SQL in dbms
 
input buffering in lexical analysis in CD
input buffering in lexical analysis in CDinput buffering in lexical analysis in CD
input buffering in lexical analysis in CD
 
ASME-B31.4-2019-estandar para diseño de ductos
ASME-B31.4-2019-estandar para diseño de ductosASME-B31.4-2019-estandar para diseño de ductos
ASME-B31.4-2019-estandar para diseño de ductos
 
maths mini project ( applictions of quadratic forms and SVD ).ppt
maths mini project ( applictions of quadratic forms and SVD ).pptmaths mini project ( applictions of quadratic forms and SVD ).ppt
maths mini project ( applictions of quadratic forms and SVD ).ppt
 
Analysis and Evaluation of Dal Lake Biomass for Conversion to Fuel/Green fert...
Analysis and Evaluation of Dal Lake Biomass for Conversion to Fuel/Green fert...Analysis and Evaluation of Dal Lake Biomass for Conversion to Fuel/Green fert...
Analysis and Evaluation of Dal Lake Biomass for Conversion to Fuel/Green fert...
 
Triangulation survey (Basic Mine Surveying)_MI10412MI.pptx
Triangulation survey (Basic Mine Surveying)_MI10412MI.pptxTriangulation survey (Basic Mine Surveying)_MI10412MI.pptx
Triangulation survey (Basic Mine Surveying)_MI10412MI.pptx
 
Indian Tradition, Culture & Societies.pdf
Indian Tradition, Culture & Societies.pdfIndian Tradition, Culture & Societies.pdf
Indian Tradition, Culture & Societies.pdf
 
22CYT12 & Chemistry for Computer Systems_Unit-II-Corrosion & its Control Meth...
22CYT12 & Chemistry for Computer Systems_Unit-II-Corrosion & its Control Meth...22CYT12 & Chemistry for Computer Systems_Unit-II-Corrosion & its Control Meth...
22CYT12 & Chemistry for Computer Systems_Unit-II-Corrosion & its Control Meth...
 
Madani.store - Planning - Interview Questions
Madani.store - Planning - Interview QuestionsMadani.store - Planning - Interview Questions
Madani.store - Planning - Interview Questions
 
Overview of IS 16700:2023 (by priyansh verma)
Overview of IS 16700:2023 (by priyansh verma)Overview of IS 16700:2023 (by priyansh verma)
Overview of IS 16700:2023 (by priyansh verma)
 
priority interrupt computer organization
priority interrupt computer organizationpriority interrupt computer organization
priority interrupt computer organization
 

Power series

  • 1. POWER SERIES (TOPIC NO. 14) NAME: JAIMIN AJITKUMAR KEMKAR EN. NO.: 160123119014 BRANCH: MECHANICAL SEM. & CLASS: 4D SUBJECT: COMPLEX VARIABLES AND NUMERIC METHODS (2141905) GUIDED BY: PROF. KEYURI M. SHAH
  • 2. CONTENTS • SINGULAR POINT AND CLASSIFICATIONS • RESIDUE THEOREM • ROUCHE’S THEOREM (WITHOUT PROOF)
  • 3. SINGULAR POINTS • A POINT AT WHICH THE FUNCTION 𝑓 𝑧 IS NOT ANALYTIC IS KNOWN AS SINGULAR POINT OR SINGULARITY OF THE FUNCTION. TYPES OF SINGULARITIES 1. ISOLATED SINGULARITY A SINGULAR POINT 𝑧 = 𝑎 IS KNOWN AS AN ISOLATED SINGULARITY IF THERE IS NO OTHER SINGULARITY WITHIN A SMALL CIRCLE SURROUNDING THE POINT 𝑧 = 𝑎 ; OTHERWISE IT IS CALLED A NON- ISOLATED SINGULARITY. EX. 𝑓 𝑧 = 𝑧2 𝑧−2 HAS ISOLATED SINGULARITY AT 𝑧 = 2.
  • 4. 2. REMOVABLE SINGULARITY AN ISOLATED SINGULAR POINT 𝑧 = 𝑎 IS KNOWN AS A REMOVABLE SINGULARTITY IF lim 𝑧→𝑎 𝑓(𝑧) EXISTS OR THERE IS NO NEGATIVE POWER TERM OF (𝑧 − 𝑎) IN LAURENT’S SERIES OF 𝑓(𝑧), I.E., 𝑏 𝑛 = 0. EX. 𝑓 𝑧 = sin 𝑧 𝑧 𝑧 = 0 IS A SINGULARITY OF 𝑓(𝑧). lim 𝑧→0 sin 𝑧 𝑧 = 1 ∴ 𝑓(𝑧) HAS A REMOVABLE SINGULARITY AT 𝑧 = 0. 3. ESSENTIAL SINGULARITY A SINGULAR POINT 𝑧 = 𝑎 IS KNOWN AS AN ESSENTIAL SINGULARITY IF THE NUMBER OF NEGATIVE POWER TERMS OF (𝑧 − 𝑎) IN LAURENT’S SERIES IS INFINITE. EX. 𝑓 𝑧 = 𝑒 1 𝑧−2 = 1 + 1 𝑧−2 + 1 2! 1 (𝑧−2)2 + ⋯ SINCE THE NUMBER OF NEGATIVE POWER TERMS OF (𝑧 − 2) IS INFINITE, 𝑧 = 2 IS AN ESSENTIAL SINGULARITY.
  • 5. 4. POLE OF ORDER m AN ISOLATED SINGULARITY 𝑧 = 𝑎 IS KNOWN AS A POLE OF ORDER m IF IN LAURENT’S SERIES, ALL THE NEGATIVE POWER OF (𝑧 − 𝑎) AFTER THE 𝑚 𝑡ℎ POWER ARE ZERO, I.E., HIGHEST POWER OF 1 𝑧−𝑎 IS m. EX. 𝑓 𝑧 = 1 (𝑧−1)(𝑧−3)2 𝑧 = 1 IS A POLE OF ORDER 1 AND 𝑧 = 3 IS A POLE ORDER 2. NOTE: A POLE OF ORDER ONE IS ALSO KNOWN AS A SIMPLE POLE.
  • 6. RESIDUE • THE COEFFICIENT OF 1 𝑧−𝑎 IN THE LAURENT SERIES EXPANSION OF 𝑓(𝑧) IS KNOWN AS THE RESIDUE OF 𝑓(𝑧) AT 𝑧 = 𝑎. • THE LAURENT SERIES EXPANSION ABOUT 𝑧 = 𝑎 𝑓 𝑧 = 𝑛=0 ∞ 𝑎 𝑛(𝑧 − 𝑎) 𝑛 + 𝑛=1 ∞ 𝑏 𝑛 (𝑧−𝑎) 𝑛 RESIDUE AT (𝑧 = 𝑎)= COEFFICIENT OF 1 𝑧−𝑎 = 𝑏1 = 1 2𝜋𝑖 𝑐 𝑓(𝑧) (𝑧−𝑎)−1+1 𝑑𝑧 𝑏 𝑛 = 1 2𝜋𝑖 𝑓(𝑧) (𝑧−𝑎)−𝑛+1 𝑑𝑧 = 1 2𝜋𝑖 𝑐 𝑓 𝑧 𝑑𝑧
  • 7. EVOLUATION OF RESIDUES 1. RESIDUE AT A SIMPLE POLE i. IF 𝑓(𝑧) HAS A SIMPLE POLE AT 𝑧 = 𝑎 THEN 𝑅𝑒𝑠 𝑓 𝑧 ; 𝑧 = 𝑎 = lim 𝑧→𝑎 𝑧 − 𝑎 𝑓(𝑧) ii. IF 𝑓 𝑧 IS OF THE FORM 𝑓 𝑧 = 𝑔(𝑧) ℎ(𝑧) , WHERE 𝑔(𝑧) AND ℎ(𝑧) ARE ANALYTIC AT 𝑧 = 𝑎. IF ℎ 𝑎 = 0 BUT ℎ′(𝑎) ≠ 0 THEN 𝑧 = 𝑎 IS A SIMPLE POLE. IF ℎ 𝑎 = 0 NUT 𝑔(𝑎) ≠ 0 THEN 𝑅𝑒𝑠 𝑓 𝑧 ; 𝑧 = 𝑎 = 1 (𝑚−1)! lim 𝑧→𝑎 𝑔(𝑧) ℎ′(𝑧) 2. RESIDUE AT A POLE OF ORDER m IF 𝑓(𝑧)HAS A POLE OF ORDER m AT 𝑧 = 𝑎 THEN 𝑅𝑒𝑠 𝑓 𝑧 ; 𝑧 = 𝑎 = 1 (𝑚−1)! lim 𝑧→𝑎 𝑑 𝑚−1 𝑑𝑧 𝑚−1 𝑧 − 𝑎 𝑚 𝑓(𝑧)
  • 8. ROUCHE’S THEOREM • LET 𝑓(𝑧) AND 𝑔(𝑧) BE ANALYTIC IN A DOMAIN D AND HAVE FINITE NUMBER OF ZEROS IN D. • SUPPOSE C IS A SIMPLE CLOSED CONTOUR IN D SUCH THAT NO ZEROS OF 𝑓(𝑧) AN 𝑔(𝑧) LIE ON C AND 𝑔(𝑧) < 𝑓(𝑧) ON C. • THEN THE TOTAL NUMBER OF ZEROS OF 𝑓 𝑧 + 𝑔(𝑧) IS EQUAL TO THE TOTAL NUMBER OF ZEROS OF 𝑓(𝑧) INSIDE C. • I.E., 𝑁𝑓+𝑔 = 𝑁𝑓
  • 9. EX. USE ROUCHE’S THEOREM TO DETERMINE THE NUMBER OF ZEROS OF THE POLYNOMIAL 𝒛 𝟔 + 𝟓𝒛 𝟒 + 𝒛 𝟑 + 𝟐𝒛 INSIDE THE CIRCLE 𝒛 = 𝟏 LET 𝑓 𝑧 = 𝒛 𝟔 + 𝟓𝒛 𝟒 + 𝒛 𝟑 + 𝟐𝒛 LET 𝑓 𝑧 = −5𝑧4 , 𝑔 𝑧 = 𝑧6 + 𝑧3 + 2𝑧 , 𝐶: 𝑧 = 1 ON CIRCLE 𝐶: 𝑧 = 1 𝑓 𝑧 = −5𝑧4 = 5 And 𝑔(𝑧) = 𝑧6 + 𝑧3 − 2𝑧 ≤ 𝑧6 + 𝑧3 + 2 𝑧 = 4 ∴ 𝑔(𝑧) ≤ 4 ∴ 𝑓(𝑧) = 5 > 4 ≥ 𝑔(𝑧) ∴ 𝑓(𝑧) > 𝑔(𝑧)
  • 10. SINCE 𝑓(𝑧) AND 𝑔(𝑧) ARE ANALYTIC WITHIN AND ON C, ALSO 𝑓(𝑧) AND 𝑓 𝑧 + 𝑔(𝑧) ARE NONZERO ON C, BY ROUCHE’S THEOREM, 𝑁𝑓+𝑔 = 𝑁𝑓 𝑓 𝑧 = −5𝑧4 HAS A ZERO OF ORDER 4 INCIDE C. 𝑁𝑓 = 4 ∴ 𝑁𝑓+𝑔 = 4 HENCE, 𝑓 𝑧 HAS FOUR ZEROS INSIDE C.
  • 11. REFERANCES • BOOK: Mc Graw Hill (CVNM)