COMPLEX ANALYSIS AND APPLICATIONS.

Elective course.

 

I_ Description:

The course extends the technique of analysis to functions of complex variables with emphasis on application to engineering problems.

 

II_ Prerequisite:

Analysis 1.

 

III_ Course contents (45min/week x 14 weeks).

            1/_ Revision:

            Set. Natural, rational, real numbers.

            Product of 2 sets. Relations, domains, range and graphs.

            Mappings. Functions. Graphs of functions.

            Composite functions. Inverse functions.

 

            2/_ Revision:

            Solution of 2 equations in 2 unknowns and 2x2 matrices.

            Addition and multiplication rules of 2x2 matrices.

            The inverses of some 2x2 matrices.

            Isomorphism between real numbers and (a, 0; 0, a) matrices.

            Isomorphism between complex numbers and (a, b; -b, a) matrices.

            Multiplication rules of two complex numbers in Cartesian and in polar form.

            Application of complex numbers to analytical geometry and alternating electricity            problems.

 

            3/_ Derivatives:

            Derivative of a real function of one real variable.

            Derivative matrix of a real vector function of a real vector variables.

            The equations of continuum mechanics.

 

            4/_ Derivatives (continued):

            Derivative of a composite real vector function of a real vector variable.

            Notation for derivative of a composite real vector function of a real vector variable.

            Derivative of the inverse function of an arbitrarily given one to one and onto       function.

 

            5/_ A real two-dimensional vector function of two real variables  f : RxR --->  RxR .

            Isogonal transformations and preservation of shapes: The geometric reflection,   rotation, translation and similarity transformations. Geometric inversion.    Mercator projection. Transverse Mercator projection.

 

            6/_ Derivatives of a complex function of one complex variable:

            A complex function of a complex variables  f : C ---> C .

            Derivative of a complex function of one complex variable.

            Corresondence between the derivative matrix and a complex number.

            The Cauchy-Riemann condition.

            The mapping of a analytic (or holomorphic) function is isogonal.

            Examples of the mappings of analytic functions: z^2, exp(z), 1/z .

            The harmonicity of individual real and imaginary components of an analytic         function.

 

            7/_ Applications of conformal mapping to engineering problems (Part I):

            Heat conduction: Adding cooling fins to metallic plate, heat gradient in a thin glass          plate.

            Electrical problems: Potential problems and the flow of electrical current in a thin            metallic plate.

            Water permeation problems: Flow of ground water around and through a dam.

           

            8/_ Derivatives of a composite complex function of one complex variable:

            The partial derivatives ¶(f o g)/¶z and ¶(f o g)/¶z* of the composite function f o g            where g : CxC ---> C is defined by g(z, z*) = (x + iy).

            Determining a complex analytic function from only its real part.

            The polynomial complex functions.

            The rational complex functions.

 

            9/_ The infinite power series as a complex function:

            Theorems on the radius of convergence for a power series.

            The exponential and trigonometric function of a complex variable.

 

            10/_ The contour integration of a function.:

            Green's theorem.

            The integration of a complex function along a closed boundary.

            Cauchy's integral formula.

            Formula for higher derivatives of an analytic function.

 

            11/_ Taylor's finite series for an analytic function.

            Zeros and poles of an analytic function.

 

            12/_ Applications of conformal mapping to engineering problems (Part II):

            Schwarz-Christoffel transformation.

            The Joukowski transformation of flow around a circle into flows around a circular          arc and aerofoil of finite thickeness. Lift calculation.

            Determining the transformation from a circle to an arbitrary curve.

            The problem of plane stress in a thin metal membrane. Muskhelishvili's method.

 

            13/_ Laplace transformation of one sided functions.

            Solution to the differential equation for the method of control by feed-back in     engineering.

 

            14/_ Fourier transformation of periodic functions.

            Application to problems in physics.

 

 

IV_ References:

            1/_ Churchill: Complex variables and applications, 3rd ed.

            2/_ Nguyen Xuan Dinh: Ham phuc.

            3/_ LV Alfors: Complex analysis, McGraw-Hill, NewYork, 3rd ed, 1966.