Ingegneria Elettronica | Mathematics III

cod. 0612400003

MATHEMATICS III

	0612400003
	DIPARTIMENTO DI INGEGNERIA INDUSTRIALE
	EQF6
	ELECTRONIC ENGINEERING
	2016/2017

PERCORSO COMUNE Cohort 2015/2016

	OBBLIGATORIO
	YEAR OF COURSE 2
	YEAR OF DIDACTIC SYSTEM 2012
	PRIMO SEMESTRE

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/05	6	60	LESSONS	BASIC COMPULSORY SUBJECTS

	Objectives
	KNOWLEDGE AND UNDERSTANDING THE TEACHING AIMS AT ACQUIRING THE FOLLOWING ELEMENTS OF MATHEMATICAL ANALYSIS AND COMPLEX ANALYSIS: COMPLEX FUNCTIONS OF COMPLEX VARIABLES, FOURIER SERIES, FOURIER TRANSFORM, LAPLACE TRANSFORM, PARTIAL DIFFERENTIAL EQUATIONS. THE SPECIFIC LEARNING OUTCOMES ACTUALLY CONSIST IN THE ACHIEVEMENT OF RESULTS AND PROOF TECHNIQUES, AS WELL AS IN THE ABILITY TO SOLVE EXERCISES AND TO DEAL, IN A CONSTRUCTIVE MANNER, WITH ADVANCED TEXTBOOKS FOR A SUFFICIENTLY INDEPENDENT APPROACH TO THE PROBLEM SOLVING. THE OBJECTIVE OF PROMOTING IN THE STUDENT SUITABLE COMPREHENSION SKILLS OF THE ISSUES ADDRESSED – ENSURING AT THE SAME TIME THE ACHIEVEMENT OF AN APPROPRIATE LEVEL OF KNOWLEDGE AND FAMILIARITY WITH THE RELATED PROOF TECHNIQUES – WILL BE PURSUED BY PRESENTING THE THEORETICAL PART IN A RIGOROUS BUT CONCISE WAY AND ACCOMPANYING IT WITH PARALLEL EXERCISE SESSIONS DESIGNED TO PROMOTE A MEANINGFUL UNDERSTANDING OF THE CONCEPTS. APPLYING KNOWLEDGE AND UNDERSTANDING WITH REGARD TO THE APPLIED KNOWLEDGE AND COMPREHENSION SKILLS THAT WILL BE PROMOTED IN THE STUDENT, THE LEARNING OUTCOMES CAN BE SUMMARIZED AS FOLLOWS. BEING ABLE TO APPLY THEOREMS AND RULES IN PROBLEMS SOLVING. BEING ABLE TO SOLVE EXERCISES OF COMPLEX ANALYSIS. BEING ABLE TO COMPUTE THE FOURIER SERIES EXPANSION OF A FUNCTION. BEING ABLE TO CALCULATE FOURIER AND LAPLACE TRANSFORMS AND INVERSE LAPLACE TRANSFORMS. BEING ABLE TO SOLVE ORDINARY DIFFERENTIAL EQUATIONS, SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS AND INTEGRO-DIFFERENTIAL EQUATIONS BY USING TRANSFORMS. BEING ABLE TO SOLVE BOUNDARY VALUE PROBLEMS FOR THE LAPLACE EQUATION, THE HEAT EQUATION AND THE VIBRATING STRING EQUATION. BEING ABLE TO IDENTIFY THE BEST METHODS TO EFFICIENTLY SOLVE A MATHEMATICAL PROBLEM. BEING ABLE TO EXPLAIN THE RESOLUTION OF EXERCISES IN THE WRITTEN PROOF. BEING ABLE TO EXPLAIN VERBALLY THE LEARNED KNOWLEDGE. BEING ABLE TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE LESSONS AND TO DEEPEN THE TOPICS DEALT BY USING MATERIALS DIFFERENT FROM THE PROPOSED ONES. THE ACQUISITION OF SUCH KNOWLEDGE AND THE RELATED COMPREHENSION SKILLS, IN ADDITION TO TRAINING THE STUDENT TO A RIGOROUS APPROACH TO PROBLEMS SOLVING IN PROFESSIONAL CONTEXTS, WILL GUARANTEE THE ACHIEVEMENT OF AN ADEQUATE LEVEL OF COMPETENCE EVEN TO UNDERTAKE FURTHER STUDY WITH A SUFFICIENT DEGREE OF MATURITY AND AUTONOMY.

KNOWLEDGE AND UNDERSTANDING
THE TEACHING AIMS AT ACQUIRING THE FOLLOWING ELEMENTS OF MATHEMATICAL ANALYSIS AND COMPLEX ANALYSIS: COMPLEX FUNCTIONS OF COMPLEX VARIABLES, FOURIER SERIES, FOURIER TRANSFORM, LAPLACE TRANSFORM, PARTIAL DIFFERENTIAL EQUATIONS. THE SPECIFIC LEARNING OUTCOMES ACTUALLY CONSIST IN THE ACHIEVEMENT OF RESULTS AND PROOF TECHNIQUES, AS WELL AS IN THE ABILITY TO SOLVE EXERCISES AND TO DEAL, IN A CONSTRUCTIVE MANNER, WITH ADVANCED TEXTBOOKS FOR A SUFFICIENTLY INDEPENDENT APPROACH TO THE PROBLEM SOLVING.
THE OBJECTIVE OF PROMOTING IN THE STUDENT SUITABLE COMPREHENSION SKILLS OF THE ISSUES ADDRESSED – ENSURING AT THE SAME TIME THE ACHIEVEMENT OF AN APPROPRIATE LEVEL OF KNOWLEDGE AND FAMILIARITY WITH THE RELATED PROOF TECHNIQUES – WILL BE PURSUED BY PRESENTING THE THEORETICAL PART IN A RIGOROUS BUT CONCISE WAY AND ACCOMPANYING IT WITH PARALLEL EXERCISE SESSIONS DESIGNED TO PROMOTE A MEANINGFUL UNDERSTANDING OF THE CONCEPTS.

APPLYING KNOWLEDGE AND UNDERSTANDING
WITH REGARD TO THE APPLIED KNOWLEDGE AND COMPREHENSION SKILLS THAT WILL BE PROMOTED IN THE STUDENT, THE LEARNING OUTCOMES CAN BE SUMMARIZED AS FOLLOWS.
BEING ABLE TO APPLY THEOREMS AND RULES IN PROBLEMS SOLVING. BEING ABLE TO SOLVE EXERCISES OF COMPLEX ANALYSIS. BEING ABLE TO COMPUTE THE FOURIER SERIES EXPANSION OF A FUNCTION. BEING ABLE TO CALCULATE FOURIER AND LAPLACE TRANSFORMS AND INVERSE LAPLACE TRANSFORMS. BEING ABLE TO SOLVE ORDINARY DIFFERENTIAL EQUATIONS, SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS AND INTEGRO-DIFFERENTIAL EQUATIONS BY USING TRANSFORMS. BEING ABLE TO SOLVE BOUNDARY VALUE PROBLEMS FOR THE LAPLACE EQUATION, THE HEAT EQUATION AND THE VIBRATING STRING EQUATION.
BEING ABLE TO IDENTIFY THE BEST METHODS TO EFFICIENTLY SOLVE A MATHEMATICAL PROBLEM.
BEING ABLE TO EXPLAIN THE RESOLUTION OF EXERCISES IN THE WRITTEN PROOF. BEING ABLE TO EXPLAIN VERBALLY THE LEARNED KNOWLEDGE.
BEING ABLE TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE LESSONS AND TO DEEPEN THE TOPICS DEALT BY USING MATERIALS DIFFERENT FROM THE PROPOSED ONES.
THE ACQUISITION OF SUCH KNOWLEDGE AND THE RELATED COMPREHENSION SKILLS, IN ADDITION TO TRAINING THE STUDENT TO A RIGOROUS APPROACH TO PROBLEMS SOLVING IN PROFESSIONAL CONTEXTS, WILL GUARANTEE THE ACHIEVEMENT OF AN ADEQUATE LEVEL OF COMPETENCE EVEN TO UNDERTAKE FURTHER STUDY WITH A SUFFICIENT DEGREE OF MATURITY AND AUTONOMY.

	Prerequisites
	PREREQUISITES FOR A SUCCESSFUL ACHIEVEMENT OF THE PROPOSED GOALS AND, IN PARTICULAR, FOR A PROPER UNDERSTANDING OF THE CONTENTS SCHEDULED FOR THE TEACHING, STUDENTS ARE REQUIRED TO MASTER THE FOLLOWING PARTICULARLY USEFUL PREREQUISITES: KNOWLEDGE OF THE INTEGRAL CALCULUS, IN PARTICULAR INTEGRATION OF FUNCTIONS OF ONE VARIABLE, INTEGRALS ON CURVES, INTEGRALS OF DIFFERENTIAL FORMS; KNOWLEDGE OF SERIES EXPANSIONS, IN PARTICULAR NUMERICAL SERIES AND SERIES OF FUNCTIONS; KNOWLEDGE OF FUNCTIONS OF SEVERAL VARIABLES AND ORDINARY DIFFERENTIAL EQUATIONS. MANDATORY PREPARATORY TEACHINGS MATEMATICA II

Prerequisites

PREREQUISITES
FOR A SUCCESSFUL ACHIEVEMENT OF THE PROPOSED GOALS AND, IN PARTICULAR, FOR A PROPER UNDERSTANDING OF THE CONTENTS SCHEDULED FOR THE TEACHING, STUDENTS ARE REQUIRED TO MASTER THE FOLLOWING PARTICULARLY USEFUL PREREQUISITES: KNOWLEDGE OF THE INTEGRAL CALCULUS, IN PARTICULAR INTEGRATION OF FUNCTIONS OF ONE VARIABLE, INTEGRALS ON CURVES, INTEGRALS OF DIFFERENTIAL FORMS; KNOWLEDGE OF SERIES EXPANSIONS, IN PARTICULAR NUMERICAL SERIES AND SERIES OF FUNCTIONS; KNOWLEDGE OF FUNCTIONS OF SEVERAL VARIABLES AND ORDINARY DIFFERENTIAL EQUATIONS.

MANDATORY PREPARATORY TEACHINGS
MATEMATICA II

	Contents
	COMPLEX FUNCTIONS OF COMPLEX VARIABLES: HOLOMORPHIC FUNCTIONS AND THEIR PROPERTIES. THE CAUCHY-RIEMANN CONDITIONS. ELEMENTARY FUNCTIONS IN THE COMPLEX FIELD. SINGULAR POINTS. CAUCHY’S THEOREM AND CAUCHY’S INTEGRAL FORMULAS. MORERA THEOREM. MEAN VALUE THEOREM. LIOUVILLE THEOREM. TAYLOR’S AND LAURENT SERIES. CLASSIFICATION OF SINGULAR POINTS. RESIDUES, THE RESIDUE THEOREM AND ITS APPLICATION TO THE EVALUATION OF INTEGRALS OF REAL FUNCTIONS (HOURS LECTURE/EXERCISE SESSIONS 8/8) FOURIER SERIES: DEFINITIONS. EXAMPLES. BESSEL INEQUALITY. PUNCTUAL CONVERGENCE THEOREM. UNIFORM CONVERGENCE THEOREM. SERIES INTEGRATION. SERIES DERIVATION. (6/6). FOURIER TRANSFORM: DEFINITION AND PROPERTIES. THE RELATIONSHIP BETWEEN DERIVATION AND MULTIPLICATION BY MONOMIALS. CONVOLUTION TRANSFORM. INVERSION FORMULA. (4/6) LAPLACE TRANSFORM: DEFINITION AND PROPERTIES. BEHAVIOR OF LAPLACE TRANSFORM AT INFINITY. INITIAL AND FINAL VALUE THEOREM. LAPLACE TRANSFORMS OF DERIVATIVES. MULTIPLICATION BY POWERS OF T. LAPLACE TRANSFORM OF INTEGRALS. DIVISION BY T. PERIODIC FUNCTIONS. CONVOLUTION TRANSFORM. INVERSE LAPLACE TRANSFORM. INVERSION FORMULAS. LAPLACE TRANSFORM AND INVERSE LAPLACE TRANSFORM CALCULATIONS. APPLICATION TO ORDINARY DIFFERENTIAL EQUATIONS, SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS AND INTEGRO-DIFFERENTIAL EQUATIONS. (5/10) PARTIAL DIFFERENTIAL EQUATIONS: INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS (PDE). HEAT, WAVE AND LAPLACE EQUATIONS. BOUNDARY VALUE PROBLEMS. SOLUTIONS OF LINEAR PDE USING SEPARATION OF VARIABLES AND LAPLACE TRANSFORMS. (2/5) TOTAL HOURS: (25/35)

Contents

COMPLEX FUNCTIONS OF COMPLEX VARIABLES: HOLOMORPHIC FUNCTIONS AND THEIR PROPERTIES. THE CAUCHY-RIEMANN CONDITIONS. ELEMENTARY FUNCTIONS IN THE COMPLEX FIELD. SINGULAR POINTS. CAUCHY’S THEOREM AND CAUCHY’S INTEGRAL FORMULAS. MORERA THEOREM. MEAN VALUE THEOREM. LIOUVILLE THEOREM. TAYLOR’S AND LAURENT SERIES. CLASSIFICATION OF SINGULAR POINTS. RESIDUES, THE RESIDUE THEOREM AND ITS APPLICATION TO THE EVALUATION OF INTEGRALS OF REAL FUNCTIONS (HOURS LECTURE/EXERCISE SESSIONS 8/8)

FOURIER SERIES: DEFINITIONS. EXAMPLES. BESSEL INEQUALITY. PUNCTUAL CONVERGENCE THEOREM. UNIFORM CONVERGENCE THEOREM. SERIES INTEGRATION. SERIES DERIVATION. (6/6).

FOURIER TRANSFORM: DEFINITION AND PROPERTIES. THE RELATIONSHIP BETWEEN DERIVATION AND MULTIPLICATION BY MONOMIALS. CONVOLUTION TRANSFORM. INVERSION FORMULA. (4/6)

LAPLACE TRANSFORM: DEFINITION AND PROPERTIES. BEHAVIOR OF LAPLACE TRANSFORM AT INFINITY. INITIAL AND FINAL VALUE THEOREM. LAPLACE TRANSFORMS OF DERIVATIVES. MULTIPLICATION BY POWERS OF T. LAPLACE TRANSFORM OF INTEGRALS. DIVISION BY T. PERIODIC FUNCTIONS. CONVOLUTION TRANSFORM. INVERSE LAPLACE TRANSFORM. INVERSION FORMULAS. LAPLACE TRANSFORM AND INVERSE LAPLACE TRANSFORM CALCULATIONS. APPLICATION TO ORDINARY DIFFERENTIAL EQUATIONS, SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS AND INTEGRO-DIFFERENTIAL EQUATIONS. (5/10)

PARTIAL DIFFERENTIAL EQUATIONS: INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS (PDE). HEAT, WAVE AND LAPLACE EQUATIONS. BOUNDARY VALUE PROBLEMS. SOLUTIONS OF LINEAR PDE USING SEPARATION OF VARIABLES AND LAPLACE TRANSFORMS. (2/5)

TOTAL HOURS: (25/35)

	Teaching Methods
	THE TEACHING CONSISTS OF FRONTAL LECTURES FOR A TOTAL OF 25 HOURS (AMOUNTING TO 2.5 CFU) AND CLASSROOM EXERCISE SESSIONS FOR A TOTAL OF 35 HOURS (AMOUNTING TO 3.5 CFU). ATTENDANCE TO THE COURSE IS MANDATORY, AND IT IS CERTIFIABLE EXCLUSIVELY BY USING THE PERSONAL BADGE. IN ORDER TO PARTICIPATE TO THE FINAL ASSESSMENT AND TO GAIN THE CREDITS CORRESPONDING TO THE COURSE, THE STUDENT MUST HAVE ATTENDED AT LEAST 70% OF THE HOURS OF ASSISTED TEACHING ACTIVITIES.

Teaching Methods

THE TEACHING CONSISTS OF FRONTAL LECTURES FOR A TOTAL OF 25 HOURS (AMOUNTING TO 2.5 CFU) AND CLASSROOM EXERCISE SESSIONS FOR A TOTAL OF 35 HOURS (AMOUNTING TO 3.5 CFU).
ATTENDANCE TO THE COURSE IS MANDATORY, AND IT IS CERTIFIABLE EXCLUSIVELY BY USING THE PERSONAL BADGE. IN ORDER TO PARTICIPATE TO THE FINAL ASSESSMENT AND TO GAIN THE CREDITS CORRESPONDING TO THE COURSE, THE STUDENT MUST HAVE ATTENDED AT LEAST 70% OF THE HOURS OF ASSISTED TEACHING ACTIVITIES.

	Verification of learning
	WITH REGARD TO THE LEARNING OUTCOMES OF THE TEACHING, THE FINAL EXAM AIMS TO EVALUATE: THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE THEORETICAL LECTURES AND THE CLASSROOM EXERCISE SESSIONS; THE MASTERY OF THE MATHEMATICAL LANGUAGE IN WRITTEN AND ORAL TESTS; THE SKILL OF PROVING THEOREMS; THE SKILL OF SOLVING EXERCISES; THE ABILITY TO IDENTIFY AND APPLY THE BEST AND EFFICIENT METHODS IN EXERCISES SOLVING; THE ABILITY TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE LESSONS. THE EXAM NECESSARY TO ASSESS THE ACHIEVEMENT OF THE LEARNING OBJECTIVES CONSISTS IN A WRITTEN TEST, PRELIMINARY WITH RESPECT TO THE ORAL EXAMINATION, AND IN AN ORAL INTERVIEW. THE WRITTEN TEST CONSISTS IN SOLVING PROBLEMS IMPLEMENTED ON THE BASIS OF WHAT HAS BEEN PROPOSED IN THE FRAMEWORK OF THE THEORETICAL LECTURES AND EXERCISE SESSIONS. SUCH A WRITTEN TEST, THAT THE STUDENT WILL HAVE TO FACE IN TOTAL AUTONOMY, WILL LAST 3 HOURS. IN ITS EVALUATION, THE RESOLUTION METHODS WILL BE TAKEN INTO ACCOUNT TOGETHER WITH THE CLARITY AND COMPLETENESS OF EXPOSITION. IN THE CASE OF PRODUCTION OF A SUFFICIENT PROOF, IT WILL BE EVALUATED THROUGH QUALITATIVE SCALES (RANGES OF MARKS). THE INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE OF ALL THE TOPICS OF THE TEACHING, AND WILL COVER DEFINITIONS, THEOREMS PROOFS, EXERCISES SOLVING. THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY CUM LAUDE), WILL DEPEND ON THE RANGE OF MARKS OF THE WRITTEN TEST, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW.

Verification of learning

WITH REGARD TO THE LEARNING OUTCOMES OF THE TEACHING, THE FINAL EXAM AIMS TO EVALUATE: THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE THEORETICAL LECTURES AND THE CLASSROOM EXERCISE SESSIONS; THE MASTERY OF THE MATHEMATICAL LANGUAGE IN WRITTEN AND ORAL TESTS; THE SKILL OF PROVING THEOREMS; THE SKILL OF SOLVING EXERCISES; THE ABILITY TO IDENTIFY AND APPLY THE BEST AND EFFICIENT METHODS IN EXERCISES SOLVING; THE ABILITY TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE LESSONS.
THE EXAM NECESSARY TO ASSESS THE ACHIEVEMENT OF THE LEARNING OBJECTIVES CONSISTS IN A WRITTEN TEST, PRELIMINARY WITH RESPECT TO THE ORAL EXAMINATION, AND IN AN ORAL INTERVIEW.
THE WRITTEN TEST CONSISTS IN SOLVING PROBLEMS IMPLEMENTED ON THE BASIS OF WHAT HAS BEEN PROPOSED IN THE FRAMEWORK OF THE THEORETICAL LECTURES AND EXERCISE SESSIONS. SUCH A WRITTEN TEST, THAT THE STUDENT WILL HAVE TO FACE IN TOTAL AUTONOMY, WILL LAST 3 HOURS. IN ITS EVALUATION, THE RESOLUTION METHODS WILL BE TAKEN INTO ACCOUNT TOGETHER WITH THE CLARITY AND COMPLETENESS OF EXPOSITION.
IN THE CASE OF PRODUCTION OF A SUFFICIENT PROOF, IT WILL BE EVALUATED THROUGH QUALITATIVE SCALES (RANGES OF MARKS).
THE INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE OF ALL THE TOPICS OF THE TEACHING, AND WILL COVER DEFINITIONS, THEOREMS PROOFS, EXERCISES SOLVING.
THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY CUM LAUDE), WILL DEPEND ON THE RANGE OF MARKS OF THE WRITTEN TEST, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW.

	Texts
	TEXTS NEEDED FOR THE INDIVIDUAL STUDY OF THE TEACHING PROGRAM WRITTEN NOTES GIVEN BY THE TEACHER. C. D’APICE, R. MANZO, VERSO L’ESAME DI MATEMATICA III, MAGGIOLI, 2015. COMPLEMENTARY REFERENCES TEXTS MURRAY R. SPIEGEL, VARIABILI COMPLESSE, COLLANA SCHAUM’S. MURRAY R. SPIEGEL, ANALISI DI FOURIER, COLLANA SCHAUM’S. MURRAY R. SPIEGEL, TRASFORMATE DI LAPLACE, COLLANA SCHAUM’S. PAUL DUCHATEAU, D. ZACHMANN, PARTIAL DIFFERENTIAL EQUATIONS, SCHAUM’S OUTLINES SERIES.

Texts

TEXTS NEEDED FOR THE INDIVIDUAL STUDY OF THE TEACHING PROGRAM
WRITTEN NOTES GIVEN BY THE TEACHER.
C. D’APICE, R. MANZO, VERSO L’ESAME DI MATEMATICA III, MAGGIOLI, 2015.

COMPLEMENTARY REFERENCES TEXTS
MURRAY R. SPIEGEL, VARIABILI COMPLESSE, COLLANA SCHAUM’S.
MURRAY R. SPIEGEL, ANALISI DI FOURIER, COLLANA SCHAUM’S.
MURRAY R. SPIEGEL, TRASFORMATE DI LAPLACE, COLLANA SCHAUM’S.
PAUL DUCHATEAU, D. ZACHMANN, PARTIAL DIFFERENTIAL EQUATIONS, SCHAUM’S OUTLINES SERIES.