The theory of functions of one complex variable is studied in this subject. Unlike the case of the functions of one or several real variables (studied Differential and Integral Calculus I and II), the focus here is fully on the differentiable case, since complex differentiable functions have much richer properties. Some of these properties and their applications to different fields of real and complex analysis are studied.



Complex Analysis, together with Differential and Integral Calculus I and II form a module. These three subjects systematically present together the basic concepts, techniques and applications of single variable differential calculus, both real and complex, or several real variables. This module aims to allow students to acquire sufficient knowledge to enable them to understand the topics taught and apply them in various fields.



Students are required to know the general concept of differentiability taught in Differential and Integral Calculus I and II in order to follow this subject. Knowledge of the line integration along curves taught in Differential and Integral Calculus II is strongly recommended.

SPECIFIC SKILLS



M15CM18: Know the main properties of functions of complex variables. Recognize analytic functions, harmonic functions and elementary functions.

M15CM19: Assimilate the statements and applications of the different Cauchy integral theorems.

M15CM20: Expand functions in Taylor and Laurent series.

M15CM21: Know the main applications and consequences of the Residue Theorem.

M15CM22: Calculate complex integrals by the method of residues. Apply it to the calculation of improper real integrals.

M15CM23: Know the basic properties of conformal mappings and their geometric properties.



LEARNING OUTCOMES



Know the main functions of one complex variable, calculate the residues and the integrals over domains of the complex plane.

1. COMPLEX NUMBERS. COMPLEX PLANE. Binomial and exponential form of complex numbers. Operations with complex numbers. Nth roots of complex numbers. Distance in the complex plane. The extended complex plane and the stereographic projections. Sequences and series of complex numbers.



2. FUNCTIONS OF COMPLEX VARIABLE. Limits and continuity. Complex derivative. Cauchy-Riemann equations. Holomorphic functions and their properties. Harmonic functions and harmonic conjugates.



3. ELEMENTARY FUNCTIONS OF COMPLEX VARIABLE. Exponential function, logarithms and branches of the logarithm function, complex exponentiations, trigonometric functions, hyperbolic functions and their inverses.



4. COMPLEX INTEGRATION AND CAUCHY'S THEOREMS. Contour integrals along curves. Primitive functions. Cauchy Integral Theorem, Cauchy Integral Formula and Generalized Cauchy Integral Formula. Consequences of Cauchy Theorem (Morera's Theorem, Liouville's Theorem. Maximum Modulus Principle).



5. TAYLOR AND LAURENT SERIES. SINGULAR POINTS. Sequences of functions and series of functions. Power series. Taylor series. Laurent series. Classification of isolated singular points and their characterization.



6. RESIDUES AND APPLICATIONS. Residues. Residue Theorem. Methods to calculate residues. Calculation of real integrals of trigonometric functions by means of residues. Calculation of improper integrals and principal values along the real line by means of residues. Argument Principle. Rouché's Theorem.



7. CONFORMAL MAPPINGS. Geometric meaning of the modulus and the argument of the derivative. Conformal mappings. Geometric study of some mappings.

Lectures: theoretical topics will be presented, following the recommended bibliography.



Classroom practices: problems and exercises proposed to the students will be solved in class, to understand, develop and work with the topics of the lectures.



Seminars: the lecturer will propose works related to the topics of the subject to the students. The students will show the work carried out in the seminar, presenting it and arguing what has been done.

• Continuous Assessment System
• Final Assessment System
• Tools and qualification percentages:
  • See the guidelines below (%): 100

CONTINUOUS ASSESSMENT: WRITTEN EXAMS AND SEMINARS



1.- Written exams covering theory and problems: up to 80% of the final mark.



Assessment criteria:

- Precision in the reasonings and definitions.

- Correctness of mathematical language.

- Clear and correctly ordered argumentation methods, explaining the steps.

- Accuracy in the results of the exercises.



2.- Participation in seminars, individual or group works, presentations (not necessarily all possibilities): up to 50% of the final mark.



Assessment criteria:

- Correct answers and proper use of mathematical language.

- Clarity in reasoning.

- Order and precision in oral and written explanations.

- Attendance.



Withdrawal from continuous assessment may be requested up to week 9 of the academic year, by writing to the lecturer of the subject.



FINAL ASSESSMENT: written exam.



The final assessment will consist of a comprehensive exam covering the whole subject. Weight 100%.

FINAL ASSESSMENT: written exam (% 100)



Assessment criteria:



- Precision in the reasonings and definitions.

- Correctness of mathematical language.

- Clear and correctly ordered argumentation methods, explaining the steps.

- Accuracy in the results of the exercises.



Non-attendance of the exam will be sufficient to withdraw from the examination assessment.

Material distributed through the eGela platform:

* Problems
* Seminars
* Course notes

Basic bibliography

AGARWAL R. P., PERERA K., PINELAS S. An Introduction to Complex Analysis. Springer, 2011.

APARICIO E. Teoría de funciones de variable compleja. UPV-EHU, 1998.

BROWN J.W., CHURCHILL R.V. Variable compleja y aplicaciones, 7a ed. McGraw-Hill, 2007.

CONWAY J. B., Functions of One Complex Variable. Springer-Verlag, 1986.

DUOANDIKOETXEA, J., RIVAS, J. Analisi Konplexua, EHUko Argitalpen Zerbitzua, 2017.

PALKA, B.P. An introduction to Complex Function Theory. Springer-Verlag ,1991.

STEIN, E.M., SHAKARCHI, R. Complex Analysis. Princeton University Press, 2003.

VOLKOVYSKI I, LUNTS G, ARAMANOVICH I. Problemas de la teoría de funciones de Variable Compleja. MIR, 1972.

In-depth bibliography

AHLFORS L. V., Complex Variables. McGraw-Hill, 1978.
LANG S. Complex Analysis. Springer, 1999.
LEVINSON N., REDHEFFER R. M., Curso de variable compleja. Reverté, 1990.
MARSDEN J. E., HOFFMANN M. J., Basic Complex Analysis. W.H. Freeman and Co. USA, 1987.
RUDIN W., Análisis real y complejo. McGraw-Hill / Interamericana de España, 1987.
SHAKARCHI R. Problems and Solutions for Complex Analysis. Springer, 1999.

Web addresses

Some very appropriate notes by Martín Rivas (UPV/EHU):
http://tp.lc.ehu.es/documents/problemas.pdf.
An online course at:
http://math.fullerton.edu/mathews/complex.html.
There are many written courses in pdf format to be found. For example: George Cain's
(http://people.math.gatech.edu/~cain/winter99/complex.html), in English, and that of B. Cuartero and F. Ruiz (http://www.unizar.es/analisis_matematico/varcompleja/prg_varcompleja.html), in Spanish.
A course by Terry Tao:
http://www.math.ucla.edu/~tao/resource/general/132.1.00w/.
A summary of the contents on the web page Physics Forums:
https://www.physicsforums.com/insights/an-overview-of-complex-differentiation-and-integration/.
The Mathematics Stack Exchange page:
https://math.stackexchange.com.