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Lecture12 - 13 Complex Numbers and Functions Complex...

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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
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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).
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