13.3 Analytic functions (Analytiske funksjoner)

Transcription

1 13.3 Analytic functions (Analytiske funksjoner) Eugenia Malinnikova, NTNU October 3,

2 This course so far: Laplace transform: solutions of ODE, integral equations, systems of ODE Periodic functions Fourier series and transform: signal analysis PDEs: method of separation of variables, Fourier series PDEs by Fourier transform Convolution operation in ODEs and PDEs Important: Revie odd and even functions!!! 2

3 The second part of the course: Introduction to complex analysis Idea: construct basic Calculus by taking a complex valued (not real valued) variable. Applications in other fields: Electrostatic and electromagnetism Aerodynamics Fluid dynamics Computer graphics Applications ithin mathematics are numerous, to mentions fe fields: Algebra, Number theory, Statistics, Differential geometry. Complex analysis is a ne language and its study requires some patience. 3

4 Complex plane It is the usual to dimensional plane, e use ne notation for the points: z = (x, y) = x + iy = (r, θ) = re iθ, < x, y <, 0 r <, π < θ 2π Here r = z = x 2 + y 2 = z z is the absolute value of z (absoluttverdien), it is distance from the point z = (x, y) to the origin (0, 0). x = R(z) is the real part (reelldelen) and y = I(z) is the imaginary part (imaginærdelen), the complex conjugate of z (den kompleks konjugerte til z) is z = x iy. 4

5 Basic sets Circular domains We fix a complex number z 0. Then {z : z z 0 = R} is a circle of radius R centered at z 0, {z : z z 0 < R} is an open disk of radius R and center z 0, {z : z z 0 R} is a closed disk of radius R and center z 0, {z : r < z z 0 < R} is an open circular ring (annulus) of radii r < R and center z 0. Half-Planes Let z = x + iy The upper half-plane is the set of points ith y > 0 and the loer half-plane is the set here y < 0. The right half-plane is the set here x > 0, the left half-plane is here x < 0. 5

6 Point sets (Punktmengder): vocabulary Let S be a set of points on the complex plane. S is called open (åpen) if for each point z S there is a disk centered at z hich is contained in S S is called linearly connected (sammenhengende) if for any to points z 1 and z 2 in S there is a continuous curve γ ith end-points z 1 and z 2 hich is contained in S ( a continuous curve is a continuous mapping γ : [0, 1] C) S is called a domain (omegn) if S is open and linearly connected. 6

7 Continuous functions Let D be a domain in C, consider a function f : D C. It is called continuous at point z 0 if for any ɛ > 0 there exists δ > 0 such that if z z 0 < δ then z D and f (z) f (z 0 ) < ɛ. In other ords, f is continuous at z 0 if lim z z 0 f (z) = f (z 0 ). Let f (z) = u(z) + iv(z), here u, v : D R. Then f is continuous at the point z 0 = (x 0, y 0 ) if and only if u and v are continuous at this point. 7

8 Examples of continuous functions f (z) = z is continuous everyhere, f (z) = Arg(z) is discontinuous at points z = x + 0i, x 0, f (z) = R(z) = x, f (z) = I(z) = y, f (z) = z, f (z) = z are continuous everyhere, f (z) = e z = e x e iy is continuous everyhere. Combinations of continuous functions (sums, differences, products, compositions) are continuous. Everything is as for real-valued functions of to variables. 8

9 Derivative Let f : D C be a continuous function. We say that f is differentiable at some point z 0 D if the folloing limit exists f (z) f (z 0 ) f (z 0 + ) f (z 0 ) lim = lim z z 0 z z 0 0 Remember that here is a complex number! When the limit exists it is called the derivative of f at the point z 0 and rite f (z 0 ). This is very different from the partial derivatives from Calculus 2! 9

10 Examples: Good old nes f (z) = C, then f (z) f (z 0) z z 0 = 0, the constant function is differentiable ith f (z 0 ) = 0. f (z) = z, then f (z) f (z 0) z z 0 f (z 0 ) = 1, = 1, the function is differentiable and f (z) = z 2 then f (z 0+) f (z 0 ) = 2z 0 + 2z 0 as 0, f (z 0 ) = 2z 0 f (z) = c k z k + c k 1 z k c 1 z + c 0 is a polynomial, then f is differentiable at each point and f (z) = kc k z k 1 + (k 1)c k 1 z k c 1 Sums and products of differentiable functions are differentiable and old rules for computations of derivatives apply. 10

11 Examples: Bad nes f (z+) f (z) = R() f (z) = R(z) = x, then has no limit as 0! This function is not differentiable, f (z) = z, then differentiable f (z+) f (z) = has no limit as 0, not f (z) = z 2, then f (z) = z z and f (z+) f (z) = z + z = z + z,the limit exists only hen z = 0, f (0) = 0 but f is not differentiable at z 0. f (z) = z, at hich points is it differentiable? 11