•

2 likes•1,075 views

Evaluation of integrals of the given functions along the unit circle on the complex plane. Application of the parametrization method. Evaluation of the integral of an odd function.

Report

Share

Report

Share

Mate factorización por binomio - 2º

The document provides examples of factorizing polynomials in Spanish. It includes factorizing quadratics, binomials, trinomials, and polynomials with greater degrees. It also discusses factoring polynomials with grouping and the difference of squares formula. The examples range from basic factoring to more advanced techniques involving multiple variables.

Figures

This document provides a summary of key concepts in algebra, geometry, trigonometry and their definitions. It includes formulas and properties for lines, polynomials, exponents, trig functions, triangles, circles, spheres, cones, cylinders, distance and the quadratic formula. Key topics covered are factoring, binomials, slope-intercept form, trig ratios, trig identities, trig reciprocals and the Pythagorean identities.

Hw #2

This document summarizes the solution steps to determine the reactions and bending moments for a beam with a fixed edge at X=0 and a roller edge at X=L. It involves applying boundary conditions to determine constants C1 through C4 in the polynomial expression for deflection W(x). The constants are then used to calculate the bending moment M(X) and shear force S(X) expressions, and to determine the reactions and moments at the two edges.

Presentation1

The document contains mathematical equations and identities relating trigonometric functions for various angle transformations including opposite, complementary, and coterminal angles. It shows relationships between sin, cos, tan, cot, sec, and csc functions when the angle is replaced with its opposite, complementary, or coterminal value.

Statistics formulaee

The document contains 38 statistical formulae related to topics like:
1) Measures of central tendency - arithmetic mean, median, mode
2) Measures of dispersion - range, variance, standard deviation
3) Correlation - Pearson's correlation coefficient, regression lines
4) Time series analysis - moving averages, exponential smoothing, measuring seasonal variations
The formulae provided calculations and methods for analyzing data distributions, relationships between variables, trends over time, and seasonal fluctuations.

Parameterized convolutional neural networks for aspect level classification

This document describes the architecture of a convolutional neural network (CNN) model with three components: a generator network CNNg, a discriminator network CNNt, and a sentence encoder network CNNs. CNNg produces an initial representation θg which is input to CNNt to generate θt. θt and the input text representations v are then input to CNNs to generate the final encoded representation θs. The networks involve convolutions using weight matrices w and biases b applied to windows of the input with pooling to generate the respective outputs θg, θt, and θs.

[112] 모바일 환경에서 실시간 Portrait Segmentation 구현하기

The document contains code snippets and equations related to machine learning operations including convolution, bilinear interpolation, softmax, and quantization. Key operations summarized include:
1. Convolution kernels using NEON intrinsics to load input vectors and perform depthwise convolutions on 8-bit data with 8-channel width.
2. Bilinear interpolation implemented as a nested loop over batch, height, width, and channels to calculate output values from 4 input values.
3. Quantized softmax using a precomputed lookup table to calculate softmax values from quantized inputs while accounting for the subtractive constant.

Answers sign-charts

1. (x - 2)(x + 3) is factorable and greater than or equal to 0 when both factors have the same sign.
2. (x + 3)2 is always greater than or equal to 0.
3. -x(x + 3) is less than or equal to 0 when one factor is positive and one is negative.

10CSL67 CG LAB PROGRAM 1

Program to recursively subdivide a tetrahedron to form 3D Sierpinski gasket. The number of recursive steps is to be specified by the user.

Efoom 2016

The document contains a 20 question multiple choice exam covering various topics in mathematics and geometry. For each question, there are 5 potential answer choices labeled a-e. The questions cover topics such as functions, complex numbers, limits, integrals, geometry, and probability. At the end, a key is provided indicating the correct answer for each question.

Mate tarea - 3º

The document provides a list of algebraic expressions to factorize. It contains three sections with multiple expressions each involving variables, constants, and operations like addition, subtraction, multiplication and exponents. The goal is to rewrite each expression as a product of factors.

حساب النهايات جبرياً

This document discusses a function f(x) and its values at different points. It defines f(x)=3x^2 and finds that f(1)=3, f(5)=75, and the limits of f(x) as x approaches 5, 6 from both sides are all equal to 30.

Introduction to Integral calculus

Introduction to integral calculus.
This slideshow deals with concept of integration. A complete explanation is provided that how integration can be written as summation. Area under the graph can be calculated through integration.

Maria

This document provides examples of factorizing polynomial expressions in various forms:
1) Factoring the difference of two squares
2) Factoring a perfect square trinomial
3) Factoring the difference and sum of two cubes
It demonstrates techniques for factoring polynomial expressions involving binomials, trinomials, and higher degree terms.

EJERCICIOS PARA EL EXAMEN

This document provides examples of factorizing polynomial expressions in various forms:
1) Factoring the difference of two binomials.
2) Factoring the square of a binomial expression.
3) Factoring a trinomial into two binomial factors.
4) Factoring a perfect square trinomial.
5) Factoring the difference of squares.
6) Factoring a trinomial that is a perfect square by addition and subtraction.
7) Factoring quadratic trinomials in the form x^2 + bx + c.
8) Factoring a perfect cube of binomials.
9) Factoring the difference and sum of perfect cubes.

Eq 1º grau

The document contains solutions to several equations:
1) It solves equations like 2x - 6 = -x + 15 for x, finding x = 7 for one.
2) It expresses fractional equations like 3x/2 - (x - 10)/8 = (x + 1)/4 in terms of x and solves for x = -8/9.
3) It solves equations involving multiple variables like 6m + 3(10 - 4m) = 25 + 8m for m and finds m = 5/14.

Mate factorización por binomio - 2º

Mate factorización por binomio - 2º

Figures

Figures

Hw #2

Hw #2

Presentation1

Presentation1

Statistics formulaee

Statistics formulaee

Parameterized convolutional neural networks for aspect level classification

Parameterized convolutional neural networks for aspect level classification

[112] 모바일 환경에서 실시간 Portrait Segmentation 구현하기

[112] 모바일 환경에서 실시간 Portrait Segmentation 구현하기

Answers sign-charts

Answers sign-charts

10CSL67 CG LAB PROGRAM 1

10CSL67 CG LAB PROGRAM 1

Efoom 2016

Efoom 2016

Mate tarea - 3º

Mate tarea - 3º

حساب النهايات جبرياً

حساب النهايات جبرياً

Introduction to Integral calculus

Introduction to Integral calculus

Maria

Maria

EJERCICIOS PARA EL EXAMEN

EJERCICIOS PARA EL EXAMEN

Eq 1º grau

Eq 1º grau

Conformal mapping

This document discusses conformal mapping, which are transformations that preserve both the magnitude and orientation of angles between curves. It provides examples of conformal mappings, including the exponential function f(z) = ez, which is conformal at every point, and the sine function f(z) = sin z, which is conformal everywhere except at odd multiples of pi/2. It also gives examples of non-conformal mappings, such as the identity function f(z) = z, which only preserves magnitude of angles.

Cálculo ii howard anton - capítulo 16 [tópicos do cálculo vetorial]

This document contains a chapter from a textbook on vector calculus. It includes 33 multi-part exercises involving concepts like divergence, curl, line integrals, and parameterizing curves. The exercises provide calculations and proofs related to vector fields and vector operations in three dimensions.

T.I.M.E. JEE Advanced 2013 Solution Paper1

This document provides model solutions to questions from the JEE Advanced 2013 exam. It includes solutions to multiple choice and numerical questions across physics, chemistry and mathematics. Some key details include:
- Solutions to 20 multiple choice questions from Paper I Section I and II of the exam.
- Solutions involving concepts like circular motion, momentum, electrostatics, thermodynamics and more.
- Solutions to 4 numerical questions involving calculations related to decay of particles, fundamental frequency of a vibrating string, and more.
- Solutions to 20 multiple choice questions from Paper II Section I, II and III covering topics in chemistry, organic chemistry, coordination chemistry and biochemistry.
- Solutions to 4 numerical questions related

Gate ee 2012 with solutions

This document contains 25 multiple choice questions from a GATE EE exam, along with explanations for each answer. The questions cover topics such as probability, complex variables, circuit analysis, differential equations, and electric machines. For each question, the correct answer is identified and a brief explanation of the solution steps is provided.

Class XII CBSE Mathematics Sample question paper with solution

Class XII CBSE Mathematics Sample question paper with solutionPratima Nayak ,Kendriya Vidyalaya Sangathan

The document is a sample question paper for Class XII Mathematics. It consists of 3 sections - Section A has 10 one-mark questions, Section B has 12 four-mark questions, and Section C has 7 six-mark questions. All questions are compulsory. The paper tests concepts related to matrices, trigonometry, calculus, differential equations, and vectors. Internal choices are provided in some questions. Calculators are not permitted.Figures

This document provides mathematical formulas and definitions for topics in algebra, geometry, trigonometry, and other areas of mathematics. It includes 3 or fewer sentences summarizing key information about triangles, circles, factoring polynomials, exponents, trigonometric functions, and other concepts. Diagrams illustrate formulas for areas of geometric shapes, trigonometric functions, and other visual representations. Tables list trigonometric function values at common angles in both radians and degrees.

mathematics question bank for engineering students

This document contains questions from 5 units of a mathematics exam for an automotive engineering course. The questions cover topics like differential equations, vector calculus, complex variables, and Laplace transforms. Some sample questions include solving differential equations using variation of parameters, finding divergence and curl of vector fields, determining if functions are analytic or harmonic, and calculating Laplace transforms. The document provides 10 multiple choice questions for each of the 5 units to total 50 questions for the exam.

Math06reviewsheet (3)

This document provides a review sheet with multiple math problems involving algebra, trigonometry, and calculus concepts. There are 23 problems covering topics such as simplifying rational expressions, solving equations, graphing functions, working with complex numbers, trigonometric identities, and applying trigonometric functions. Answers are provided for selected problems. The review sheet is intended to help students practice and review a wide range of mathematics topics.

2 complex numbers part 2 of 3

Successful people replace words like "wish", "try", and "should" with "I will" in their speech and thinking. Ineffective people do not make this replacement and continue using weaker language. The document provides mathematical formulas and properties involving complex numbers, including formulas for roots of unity, sums of trigonometric series, representations of lines and circles using complex numbers, and other identities.

Solution Manual : Chapter - 01 Functions

This document contains exercises involving functions and graphs. There are multiple choice and free response questions testing understanding of properties of functions including domain, range, intercepts, maxima/minima, and graphing functions. Solutions are provided for checking work.

Conformal mapping

Conformal mapping

Cálculo ii howard anton - capítulo 16 [tópicos do cálculo vetorial]

Cálculo ii howard anton - capítulo 16 [tópicos do cálculo vetorial]

T.I.M.E. JEE Advanced 2013 Solution Paper1

T.I.M.E. JEE Advanced 2013 Solution Paper1

Gate ee 2012 with solutions

Gate ee 2012 with solutions

Class XII CBSE Mathematics Sample question paper with solution

Class XII CBSE Mathematics Sample question paper with solution

Figures

Figures

mathematics question bank for engineering students

mathematics question bank for engineering students

Math06reviewsheet (3)

Math06reviewsheet (3)

2 complex numbers part 2 of 3

2 complex numbers part 2 of 3

Solution Manual : Chapter - 01 Functions

Solution Manual : Chapter - 01 Functions

Ptolemy's theorem visualisation. 3D graphics.

Ptolemy's theorem states the following: a convex quadrilateral can be inscribed in a circle if and only if the product of the lengths of one pair of opposite sides added to the product of the lengths of the other pair is equal to the product of the lengths of the diagonals. Thus, in a cyclic quadrilateral ABCD we have
AB*DC + AD*BC = AC*BD

An inequality painted on the vase body.

The 3D picture shows an attempt at combining applied art with pure mathematics. The vase shown in the picture is an example of ceramic vessel that could exist in real world.

Still life with vases - a 3D visualisation.

The picture presenting a still life with two vases made of black polished ceramics, standing on blocks of granite and limestone, placed in an empty room illuminated by sunlight going through a big window.

How to draw arcs of huge circles - usage of the "drawing tool".

The slide explaining visually the procedure of drawing huge circles. A theoretical background of the method has been described in another slide.

How to draw arcs of huge circles - description.

In case you are interested in drawing circles with a huge diameter (by "huge" I mean here really big ones, measured in kilometers or so) there is a tricky idea needed because the rope-and-the-stick method won't work for circles with a diameter bigger than several dozen meters. Here you can use a method based on the well known fact from geometry stating that all angles inscribed in a circle and subtended by the same chord (lying on the same side of the chord) are equal.

Calculation of the volume of a bottle partially filled with a fluid.

How to calculate the volume of a flat-bottomed, corked bottle, partially filled with a fluid having only the ruler as a measuring tool?
The 3D graphics made by me and presented below gives an answer to this question.

Permutation theorem and its use to proving inequalities.

The permutation theorem is very useful when dealing with inequalities between sums of products of the two real number sequences. In numerous cases inequalities otherwise difficult to prove can be proven almost automatically.

The sum of the triangle sides lengths reciprocals vs a cyclic sum of a specif...

Proof of the inequality between the sum of the reciprocals of a triangle sides lengths and a cyclic sum of a specific form. Use of a transformed inequality between the arithmetic mean and the harmonic mean.

Indescribable numbers

1) There are a finite number of possible symbol sets that could be used to describe numbers, as the set of all symbols used by humans is finite.
2) While new symbols can be invented, the set of all possible symbol combinations remains finite.
3) Any description system based on a finite set of symbols can only describe countably many numbers, whereas the set of all real numbers is uncountably large. Therefore, there will always be real numbers that cannot be described.

Late Spring 2015 Photos

Photos taken in May and June 2015.

Cube root

A mathematical explanation of a simple trick that shows how to quickly calculate the cube root of a maximally 6-digit number that was given by a spectator. The spectator chooses a natural two-digit number, keeps it in secret and only information he/she gives publicly is its cube. Conjurer's task is to quickly guess the original number.

Ptolemy's theorem visualisation. 3D graphics.

Ptolemy's theorem visualisation. 3D graphics.

An inequality painted on the vase body.

An inequality painted on the vase body.

Still life with vases - a 3D visualisation.

Still life with vases - a 3D visualisation.

How to draw arcs of huge circles - usage of the "drawing tool".

How to draw arcs of huge circles - usage of the "drawing tool".

How to draw arcs of huge circles - description.

How to draw arcs of huge circles - description.

Calculation of the volume of a bottle partially filled with a fluid.

Calculation of the volume of a bottle partially filled with a fluid.

Permutation theorem and its use to proving inequalities.

Permutation theorem and its use to proving inequalities.

The sum of the triangle sides lengths reciprocals vs a cyclic sum of a specif...

The sum of the triangle sides lengths reciprocals vs a cyclic sum of a specif...

Indescribable numbers

Indescribable numbers

Late Spring 2015 Photos

Late Spring 2015 Photos

Cube root

Cube root

What is Rescue Session in Odoo 17 POS - Odoo 17 Slides

In this slide, we will discuss the rescue session feature in Odoo 17 Point of Sale (POS). Odoo POS allows us to manage our sales both online and offline. The rescue session helps us recover data in case of internet connectivity issues or accidental session closure.

Codeavour 5.0 International Impact Report - The Biggest International AI, Cod...

Codeavour 5.0 International Impact Report - The Biggest International AI, Cod...Codeavour International

Unlocking potential across borders! 🌍✨ Discover the transformative journey of Codeavour 5.0 International, where young innovators from over 60 countries converged to pioneer solutions in AI, Coding, Robotics, and AR-VR. Through hands-on learning and mentorship, 57 teams emerged victorious, showcasing projects aligned with UN SDGs. 🚀
Codeavour 5.0 International empowered students from 800 schools worldwide to tackle pressing global challenges, from bustling cities to remote villages. With participation exceeding 5,000 students, this year's competition fostered creativity and critical thinking among the next generation of changemakers. Projects ranged from AI-driven healthcare innovations to sustainable agriculture solutions, each addressing local and global issues with technological prowess.
The journey began with a collective vision to harness technology for social good, as students collaborated across continents, guided by mentors and educators dedicated to nurturing their potential. Witnessing the impact firsthand, teams hailing from diverse backgrounds united to code for a better future, demonstrating the power of innovation in driving positive change.
As Codeavour continues to expand its global footprint, it not only celebrates technological innovation but also cultivates a spirit of collaboration and compassion. These young minds are not just coding; they are reshaping our world with creativity and resilience, laying the groundwork for a sustainable and inclusive future. Together, they inspire us to believe in the limitless possibilities of innovation and the profound impact of young voices united by a common goal.
Read the full impact report to learn more about the Codeavour 5.0 International.How to Manage Early Receipt Printing in Odoo 17 POS

This slide will represent how to manage the early receipt printing option in Odoo 17 POS. Early receipts offer transparency and clarity for each customer regarding their individual order. Also printing receipts as orders are placed, we can potentially expedite the checkout process when the bill is settled.

Our Guide to the July 2024 USPS® Rate Change

Postal Advocate manages the mailing and shipping spends for some of the largest organizations in North America. At this session, we discussed the USPS® July 2024 rate change. Postal Advocate shared all the important information you need to know for this coming rate change that goes into effect on Sunday, July 14, 2024.
We Covered:
-What rates are changing
-How this impacts you
-What you need to do
-Savings tips

C Interview Questions PDF By Scholarhat.pdf

C Interview Questions PDF By Scholarhat

How to Create & Publish a Blog in Odoo 17 Website

A blog is a platform for sharing articles and information. In Odoo 17, we can effortlessly create and publish our own blogs using the blog menu. This presentation provides a comprehensive guide to creating and publishing a blog on your Odoo 17 website.

2 Post harvest Physiology of Horticulture produce.pptx

Post harvest Physiology of Horticulture produce

Power of Ignored Skills: Change the Way You Think and Decide by Manoj Tripathi

Power of Ignored Skills: Change the Way You Think and Decide by Manoj Tripathi Book Summary

How To Update One2many Field From OnChange of Field in Odoo 17

There can be chances when we need to update a One2many field when we change the value of any other fields in the form view of a record. In Odoo, we can do this. Let’s go with an example.

How to Manage Large Scrollbar in Odoo 17 POS

Scroll bar is actually a graphical element mainly seen on computer screens. It is mainly used to optimize the touch screens and improve the visibility. In POS there is an option for large scroll bars to navigate to the list of items. This slide will show how to manage large scroll bars in Odoo 17.

How to Manage Access Rights & User Types in Odoo 17

In Odoo, who have access to the database they are called users. There are different types of users in odoo and they have different accesses into the database. Access rights are permissions that can be set for the individual or group of users. This slide will show How to Manage Access Rights & User Types in Odoo 17.

NAEYC Code of Ethical Conduct Resource Book

NAEYC Code of Ethical Conduct Book

FINAL MATATAG Kindergarten CG 2023 pdf

FINAL MATATAG Kindergarten CG 2023.pdf

Introduction to Google Productivity Tools for Office and Personal Use

Introduction to Google Productivity Tools for Office and Personal UseExcellence Foundation for South Sudan

This is an introduction to Google Productivity Tools for office and personal use in a Your Skill Boost Masterclass by the Excellence Foundation for South Sudan on Saturday 13 and Sunday 14 July 2024. The PDF talks about various Google services like Google search, Google maps, Android OS, YouTube, and desktop applications.E-learning Odoo 17 New features - Odoo 17 Slides

Now we can take a look into the new features of E-learning module through this slide.

10th Social Studies Enrichment Material (Abhyasa Deepika) EM.pdf

10th Social Studies

RDBMS Lecture Notes Unit4 chapter12 VIEW

Description:
Welcome to the comprehensive guide on Relational Database Management System (RDBMS) concepts, tailored for final year B.Sc. Computer Science students affiliated with Alagappa University. This document covers fundamental principles and advanced topics in RDBMS, offering a structured approach to understanding databases in the context of modern computing. PDF content is prepared from the text book Learn Oracle 8I by JOSE A RAMALHO.
Key Topics Covered:
Main Topic : VIEW
Sub-Topic :
View Definition, Advantages and disadvantages, View Creation Syntax, View creation based on single table, view creation based on multiple table, Deleting View and View the definition of view
Target Audience:
Final year B.Sc. Computer Science students at Alagappa University seeking a solid foundation in RDBMS principles for academic and practical applications.
Previous Slides Link:
1. Data Integrity, Index, TAble Creation and maintenance https://www.slideshare.net/slideshow/lecture_notes_unit4_chapter_8_9_10_rdbms-for-the-students-affiliated-by-alagappa-university/270123800
2. Sequences : https://www.slideshare.net/slideshow/sequnces-lecture_notes_unit4_chapter11_sequence/270134792
About the Author:
Dr. S. Murugan is Associate Professor at Alagappa Government Arts College, Karaikudi. With 23 years of teaching experience in the field of Computer Science, Dr. S. Murugan has a passion for simplifying complex concepts in database management.
Disclaimer:
This document is intended for educational purposes only. The content presented here reflects the author’s understanding in the field of RDBMS as of 2024.

BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH LỚP 12 - GLOBAL SUCCESS - FORM MỚI 2025 - ...

BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH LỚP 12 - GLOBAL SUCCESS - FORM MỚI 2025 - ...Nguyen Thanh Tu Collection

https://app.box.com/s/ht97vau50d2v85ukt5tmi2e54f1wos2tGenetics Teaching Plan: Dr.Kshirsagar R.V.

A good teaching plan is a comprehensive write-up of the step-by-step and teaching methods helps students for understand the topic

Allopathic M1 Srudent Orientation Powerpoint

Allopathic Medical M1 Orientation

What is Rescue Session in Odoo 17 POS - Odoo 17 Slides

What is Rescue Session in Odoo 17 POS - Odoo 17 Slides

Codeavour 5.0 International Impact Report - The Biggest International AI, Cod...

Codeavour 5.0 International Impact Report - The Biggest International AI, Cod...

How to Manage Early Receipt Printing in Odoo 17 POS

How to Manage Early Receipt Printing in Odoo 17 POS

Our Guide to the July 2024 USPS® Rate Change

Our Guide to the July 2024 USPS® Rate Change

C Interview Questions PDF By Scholarhat.pdf

C Interview Questions PDF By Scholarhat.pdf

How to Create & Publish a Blog in Odoo 17 Website

How to Create & Publish a Blog in Odoo 17 Website

2 Post harvest Physiology of Horticulture produce.pptx

2 Post harvest Physiology of Horticulture produce.pptx

Power of Ignored Skills: Change the Way You Think and Decide by Manoj Tripathi

Power of Ignored Skills: Change the Way You Think and Decide by Manoj Tripathi

How To Update One2many Field From OnChange of Field in Odoo 17

How To Update One2many Field From OnChange of Field in Odoo 17

How to Manage Large Scrollbar in Odoo 17 POS

How to Manage Large Scrollbar in Odoo 17 POS

How to Manage Access Rights & User Types in Odoo 17

How to Manage Access Rights & User Types in Odoo 17

NAEYC Code of Ethical Conduct Resource Book

NAEYC Code of Ethical Conduct Resource Book

FINAL MATATAG Kindergarten CG 2023 pdf

FINAL MATATAG Kindergarten CG 2023 pdf

Introduction to Google Productivity Tools for Office and Personal Use

Introduction to Google Productivity Tools for Office and Personal Use

E-learning Odoo 17 New features - Odoo 17 Slides

E-learning Odoo 17 New features - Odoo 17 Slides

10th Social Studies Enrichment Material (Abhyasa Deepika) EM.pdf

10th Social Studies Enrichment Material (Abhyasa Deepika) EM.pdf

RDBMS Lecture Notes Unit4 chapter12 VIEW

RDBMS Lecture Notes Unit4 chapter12 VIEW

BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH LỚP 12 - GLOBAL SUCCESS - FORM MỚI 2025 - ...

BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH LỚP 12 - GLOBAL SUCCESS - FORM MỚI 2025 - ...

Genetics Teaching Plan: Dr.Kshirsagar R.V.

Genetics Teaching Plan: Dr.Kshirsagar R.V.

Allopathic M1 Srudent Orientation Powerpoint

Allopathic M1 Srudent Orientation Powerpoint

- 1. Show that C f(z)dz = 0, where f is the given function and C is the unit circle |z| = 1. a) f(z) = z3 − 1 + 3i b) f(z) = z2 + 1 z − 4 c) f(z) = sin z (z2 − 25) (z2 + 9) Solution by Mikołaj Hajduk: The points a) and b) can be solved with use of parametrization z(t) = eit where t ∈ [0, 2π] and dz = ieit dt. Ad. a) C f(z)dz = C (z3 − 1 + 3i)dz = 2π 0 ((eit )3 − 1 + 3i)ieit dt = = 2π 0 e3it ieit + (−1 + 3i)ieit dt = 2π 0 ie4it dt + 2π 0 (−1 + 3i)ieit dt = = i 2π 0 e4it dt + (−1 + 3i)i 2π 0 eit dt = i 1 4i e4it 2π 0 + (−1 + 3i)i 1 i eit 2π 0 = = 1 4 e4it 2π 0 + (−1 + 3i)eit 2π 0 = 1 4 (e4i∗2π − e4i∗0 ) + (−1 + 3i)(ei∗2π − ei∗0 ) = c 2015/09/17 02:27:23, Mikołaj Hajduk 1 / 4 next
- 2. = 1 4 ((e2iπ )4 − e0 ) + (−1 + 3i)(e2iπ − e0 ) = 1 4 (1 − 1) + (−1 + 3i)(1 − 1) = 0 Ad. b) C f(z)dz = C z2 + 1 z − 4 dz = 2π 0 (eit )2 + 1 eit − 4 ieit dt = = 2π 0 ie2it eit dt + 2π 0 ieit eit − 4 dt = i 2π 0 e3it dt + 2π 0 (eit − 4) eit − 4 dt = = i 1 3i e3it 2π 0 + ln |eit − 4| 2π 0 = 1 3 (e3i∗2π − e3i∗0 ) + (ln |ei∗2π − 4| − ln |ei∗0 − 4|) = = 1 3 ((e2iπ )3 − e0 ) + (ln |1 − 4| − ln |1 − 4|) = 1 3 (1 − 1) + (ln 3 − ln 3) = 0 Ad. c) Let’s notice that the function f is odd in its domain because sin(−z) def = ei(−z) − e−i(−z) 2i = e−iz − eiz 2i = − eiz − e−iz 2i def = − sin(z) and hence f(−z) = sin(−z) ((−z)2 − 25) ((−z)2 + 9) = − sin z (z2 − 25) (z2 + 9) = −f(z) Let CU and CB denote the circle arcs corresponding to the upper and bottom halves of the circle C. We have then C f(z)dz = CU f(z)dz + CB f(z)dz c 2015/09/17 02:27:23, Mikołaj Hajduk 2 / 4 next
- 3. The function h : CU −→ CB deﬁned as h(z) = −z is a bijection between points of the arcs CU and CB that transforms the beginning point of the arc CU into the beginning point of the arc CB and the ending point of the arc CU into the ending point of the arc CB. The arc CB is an image of the arc CU under the function h: CB = h(CU) c 2015/09/17 02:27:23, Mikołaj Hajduk 3 / 4 next
- 4. therefore, bearing in mind that f(z) is odd and h−1 (z) = −z, we get CB f(z)dz = h(CU) f(z)dz = CU f(h−1 (z))dz = CU f(−z)dz = CU −f(z)dz = − CU f(z)dz Hence C f(z)dz = CU f(z)dz + CB f(z)dz = CU f(z)dz − CU f(z)dz = 0 c 2015/09/17 02:27:23, Mikołaj Hajduk 4 / 4