{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture12

# Lecture12 - 13 Complex Numbers and Functions Complex...

This preview shows pages 1–3. Sign up to view the full content.

13 Complex Numbers and Functions Complex analysis is the theory of functions of a complex variable. Holomor- phic functions are of particular interest; these are complex differentiable in every point z U in a domain U C . If f = u + iv is holomorphic, then the real and imaginary parts u, v , understood as functions of two real vari- ables ( x, y ) R 2 , satisfy the Cauchy-Riemann equations, which is a system of two first-order partial differential equations. If, furthermore, u and v have continuous second partial derivatives, they are harmonic functions and thus solutions to Laplace’s differential equation (cf. Chapter 12) in two space di- mensions. Therefore, potential theory in 2D (to be treated in Chapter 18) is related to complex analysis, and this shall be the main motivation for its study in this lecture. There are other useful results from complex analysis, such as that Fourier series may be written down in a somewhat simpler way by using complex exponentials, or that certain complicated integrals can be evaluated by the elegant method of complex integration, the calculus of residues (Chapter 16). 13.1 Complex Numbers. Complex Plane. We follow the Cayley-Dickson construction to obtain the field of complex numbers, C , from the field of real numbers, R . Ordered pairs of real numbers We consider ordered pairs of real num- bers, ( x, y ) R × R . (432) Addition We define the addition + of ordered pairs componentwise, i. e. by ( x 1 , y 1 ) + ( x 2 , y 2 ) := ( x 1 + x 2 , y 1 + y 2 ) . (433) The set of ordered pairs together with the addition just defined forms a commutative group: 1. Closure is obvious from the definition. 78

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2. Associativity: (( x 1 , y 1 ) + ( x 2 , y 2 )) + ( x 3 , y 3 ) = ( x 1 + x 2 , y 1 + y 2 ) + ( x 3 , y 3 )(434) = ( x 1 + x 2 + x 3 , y 1 + y 2 + y 3 )(435) = ( x 1 , y 1 ) + ( x 2 + x 3 , y 2 + y 3 )(436) = ( x 1 , y 1 ) + (( x 2 , y 2 ) + ( x 3 , y 3 )) (437) 3. Identity element: the pair (0 , 0) R × R is the additive identity: (0 , 0) + ( x, y ) = ( x, y ) = ( x, y ) + (0 , 0) . (438) 4. Inverse element: for a pair ( x, y ) R × R , the additive inverse is given by ( x, y ) R × R : ( x, y ) + ( x, y ) = (0 , 0) = ( x, y ) + ( x, y ) . (439) 5. Commutativity: ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( x 1 + x 2 , y 1 + y 2 ) = ( x 2 + x 1 , y 2 + y 1 )(440) = ( x 2 , y 2 ) + ( x 1 , y 1 ) . (441) Remark: If we add a scalar product λ ( x, y ) := ( λx, λy ), λ R , we obtain a vector space, which is of course R 2 with the standard basis (1 , 0), (0 , 1).
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern