COURSE LANGUAGE: English

YEAR OF THE DEGREE PROGRAMME (I, II, III): II

SEMESTER (I, II, ANNUAL): I

CFU: 9

REQUIRED PRELIMINARY COURSES (IF MENTIONED IN THE COURSE STRUCTURE “REGOLAMENTO”) 
Calculus II – Geometry and linear Algebra 

PREREQUISITES (IF APPLICABLE) 
None.

LEARNING GOALS 
To provide the fundamental concepts and results, in view of applications, related to the theory of analytic functions, distributions, Fourier series, Fourier and Laplace transforms and their applications. 

EXPECTED LEARNING OUTCOMES (DUBLIN DESCRIPTORS) 

Knowledge and understanding 
The student will have to demonstrate knowledge of the notions (definitions, statements, proofs if provided by the program) related to the theory of holomorphic functions and integration in a complex field, distributions, Fourier series, Fourier and Laplace transforms and developed calculation tools, and knowing how to understand related topics by elaborating the acquired notions.  

Applying knowledge and understanding 
The student must demonstrate that he knows how to apply what he has learned in solving verification exercises developed by the teacher, in principle related to topics such as: calculation of integrals in the real field and in the complex field with residue theory, linear difference equations, Fourier series and transforms of periodic signals, Laplace transforms of functions and applications to linear differential problems, distributional calculus. 

COURSE CONTENT/SYLLABUS 

• Complex numbers.  
  Algebraic, trigonometric, exponential form. Properties of the modulus. Formulas of De Moivre and of the nth roots. Elementary functions in the field of complex numbers: exponential, sine and cosine, hyperbolic sine and cosine, logarithm, powers. Sequences and series in the field of complex numbers. Power series: radius of convergence and properties, term-to-term derivation. 
• Analytical functions.  
  Holomorphy and Cauchy-Riemann conditions. Line integrals of functions of complex variable. Cauchy's Theorem and Formula. Taylor series. Laurent series. Zeros of analytic functions and principles of identity. Classification of isolated singularities. Liouville's theorem. 
• Integration.  
  Lebesgue measure and Lebesgue integral. Summable functions. Passage to the limit under the integral sign. Integrals in the sense of the principal value according to Cauchy. Spaces of summable functions. 
• Residue.  
  Residue theorem. Calculation of residues in poles. Computation of integrals with the residue method. Jordan lemmas. Simple fractions. 
• Difference equations.  
  Z-transform: definition and properties. Z-anti-transform. Sequences defined by recurrence. 
• Laplace transform. 
  Signals. General informations on signals. Periodic signals. Convolution. Definition and domain of the two-sided Laplace transform. Analyticity and behavior at infinity. Notable examples of the Laplace transform. Formal properties of the Laplace transform. Laplace unilateral transform and properties. Initial and final value theorems. Antitransform. Use of the Laplace transform in linear differential models. 
• Fourier series. 
  Banach and Hilbert spaces. Energy of a periodic signal. Trigonometric polynomials. Exponential and trigonometric Fourier series. Convergence in pointwise sense and in the sense of energy 
• Fourier transform. 
  Definition of Fourier transform. Formal properties of the Fourier transform. Anti-transform. The Fourier transform and the heat equation. 
• Distributions.  
  Linear functionals. Limits in the sense of distributions. Derivative in the sense of distributions. Rules of derivation. Notable examples: δ of Dirac, v.p. 1 / t. Convolution of distributions. Space of rapidly decreasing functions and related topology. Temperate distributions and slow-growing functions. Fourier transform of temperate distributions. Laplace transform of distributions. Fourier transform of Dirac's δ, of the pulse train. Fourier transform of periodic signals. 
• Boundary value problems  
  Self-adjoint equations. Green's function, the alternative theorem. The Sturm-Liouville problem, orthogonality and eigenfunctions. 
• Partial differential equations.  
  Laplace and Poisson equations, harmonic functions, Dirichlet and Neumann problems. Solution of the Dirichlet problem for the Laplace equation in a circle. Equation of heat, Cauchy problem in the half plane. Wave equation, Cauchy problem in the half plane, mixed problem in the half strip. 

READINGS/BIBLIOGRAPHY 
Check the teacher website.

TEACHING METHODS 
The lessons will be face to face, and about one third of the lessons will be of exercises. 

EXAMINATION/EVALUATION CRITERIA 

Exam type:

• Written and oral.

In case of a written exam, questions refer to:

• Numerical exercises.
• Open answers.
• Multiple choice answers.

Evaluation criteria:
The grade is formulated by the Examination Commission based on the outcome of written test, in particular on the basis of the consistency and accuracy of the exercises performed and the adequacy of the answers provided by the student to the theory questions. The final grade is also suitably motivated to the student.