Appendix 2 Complex Analysis

Transcription

1 Appendix Complex Analsis This section is intended to review several important theorems and definitions from complex analsis. Analtic function Let f be a complex variable function of a complex value, i.e. f : The derivative of f at = x+ iis defined b f ( ) ' = lim h ( + ) ( ) f h f h where h=δ x+ iδ is a complex number. Note the difference between this definition of derivative and the definition of derivative for real-valued functions. Namel, in the real case, the limit is from left and right onl. But in the complex plan h can approach ero from various direction as show in the figure. Definition: A function f ( ) defined on an open set D is said to be analtic on D, if f '( ) exists and is continuous for all in D. Moreover, if ( ) we sa f ( ) is entire. Example: Here are some examples of analtic functions.. f ( ) = is analtic and f '( ) =.. For an polnomial n n f ( ) = an + an + + a + a is analtic, and we have n- n- f '( ) = nan + ( n-) an + + a + a 3. The function e is define b It is eas to show that - x+ i x e = e = e (cos + isin ). e is analtic and ( e )' = e f is analtic on all of, A function f ( ) can be expressed in the form of real and the imaginar parts:

2 For example, for f ( ) = ² we have ( ) = ( + ) = ( ) + ( ) f f x i u x, iv x,. where ( ) = ( + ) f f x i = ( x+ i)² = x² ² + xi ( ) iv( x ) = u x, +,. (, ) = ² ² and v( x ) u x x, = x Notice that ux = v and u = -vx These last equations are called Cauch-Riemann equation, and all analtic function satisfies them. Theorem (Cauch-Riemann criterion) The function ( ) (, ) (, ) f = u x + v x i is analtic in an open set D if and onl if the partial derivatives are continuous and satisf the Cauch-Riemann equations u = v and u = v x x In this case ( ) (, ) (, ) (, ) (, ) f = ux x + vx x i = v x u x i. The Figure illustrates the proof of the Cauch-Riemann equations. = + iδ x x = + Δx Figure Figure

3 Example Here is how the Cauch-Riemann equations are used to determine if a function is analtic.. f ( ) = ². So f ( x+ i) = ( x+ i) ² = x² ² + xi, hence ( ) and v( x, ) = x. You can easil check that u = x= v and. f ( ) =. So f ( x+ i) = x i, hence u( x, ) Contour Integral x u x, = x² ², u = = vx, and so it follows that f is analtic. = x, and v( x, ) =. You can easil check that u = = v, and so it follows that f is not analtic. x Before we can discuss integral of complex function, we need to define curves in the complex plane. Definition A complex-valued function of a real variable f ( t) continuous on [ ab, ] if. f () t exists and is continuous for all but finitel man points in ( ab ). At an discontinuous point c ( a, b). Both lim f ( t) and lim f () t and are finite. + t c 3. At the endpoints, lim f ( t) and lim f ( t) + t a t b f t is smooth or continuousl differentiable on [, ] () t are both continuous on [ ab, ]. The function () f is said to be piecewise t c exist and are finite.,. ab if f () t and exist Definition A path or a contour is a continuous piecewise smooth curve () t closed interval [ ab, ]. over a The figure illustrates a path (or contour). 3

4 Recall the definition of the Riemann integral over an interval [ ab, ]. If map [ ab, ] is defined on [ ab, ]. The contour integral of f on is the on [ ab, ]. as in the following definition. into is a path then f Riemann integral of f Definition Suppose that is a path over a closed interval [ ab, ] and that f is a continuous complex-valued function defined on the graph of. The contour integral of f on is given b b f ( d ) = f ( ( t)) '( t) dt a Definition. Let a and b be points in the region D, and and are paths in D joining a to b. is continuousl deformable into relative to D if we can continuousl move over while keeping the ends fixed without leaving D.. Let and be closed paths in D. is continuousl deformable into relative to D if we can continuousl move to overlap in both position and direction without leaving D. The figures illustrate the deformation of a path and a closed path. 4

5 Cauch's Theorem and its Consequences Cauch's theorem tells us that the integral of different path is the same as long as the paths are deformable into each other. Theorem (Cauch's Theorem) Let f be an analtic function in a region D and α and β be points in D.. If and are two paths joining α to β in D, and if is continuousl deformable into relative to D, then f ( d ) = f( d ). If and are closed paths in D, and if is continuousl deformable into relative to D, then f ( d ) = f( d ) Example Here are some examples of using Cauch's Theorem to evaluate line integrals.. Let f ( ) = ². Since f is analtic on and inside the curve in Figure below. It follows from Cauch's Theorem that f( ) d = f =. The curves and in Figure below are deformable into each other without passing through the singularit at, it follows that. Let ( ) f ( d ) = f( d ) f =. The curves and in Figure 3 below are not deformable into each other without passing through the singularit at, Cauch's theorem gives no information about the relative value of these integrals. In fact, 3. Let ( ) f ( d ) f( d ) 5

6 Im Im Im Re Re Re Figure Figure Figure 3 The consequences of Cauch s theorem help us to develop most important integral formulas. Cauch's theorem tells us that the value of an analtic function of f at point inside a curve is completel determined b its values on. This is quite a surprising propert of analtic function. Theorem (Cauch's integral formula) Suppose that f is analtic inside and on a simple closed path C with positive orientation. If is an point inside C, then f ( ζ ) f ( ) = dζ πi ζ C Theorem (Liouville's theorem) If f is entire and bounded, the f is constant. Theorem (Identit Principle) Suppose f ( ) and g( ) be analtic on a region D, and suppose that { n} is an infinite sequence of distinct points in D converging to in D. If f ( ) = g( ) for all n, then f ( ) = g( ) for all in D. n n 6