Complex Variables. Laplace Transform Z Transform. Prof. Nicolas Dobigeon. University of Toulouse IRIT/INP-ENSEEIHT

Transcription

1 Complex Variables Laplace Transform Z Transform Prof. Nicolas Dobigeon University of Toulouse IRIT/INP-ENSEEIHT nicolas.dobigeon@enseeiht.fr Prof. Nicolas Dobigeon Complex variables - LT & ZT 1 / 96

2 Outline Some Generalities Usual functions Holomorphic functions Integration and Cauchy theorem Residue theorem Laplace transform Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 2 / 96

3 Some Generalities Introduction Outline Some Generalities Introduction Limits - continuity Usual functions Holomorphic functions Integration and Cauchy theorem Residue theorem Laplace transform Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 3 / 96

4 Some Generalities Introduction Complex plane (z-plane) Complex plan is the plane equipped with the direct orthonormal basis (O; u, v). La correspondence { R 2 C (x, y) z = x + iy is bijective. By a slight abuse of notation, the point M (x, y) and its affix z = x + iy coincide. If z 0, the representation of the complex number z under the form modulus/argument is written z = ρe iθ where ( ρ ) = z = OM is the modulus z and θ = arg z is a angle measure u, OM (in rad) defined modulo 2π, i.e., ±2kπ, k Z. Prof. Nicolas Dobigeon Complex variables - LT & ZT 4 / 96

5 Some Generalities Introduction Complex function of the z-variable For any function f of complex variable { C C f : z = x + iy f (z) = P(x, y) + iq(x, y) we can define a function F : { R F : 2 R 2 (x, y) F (x, y) = (P(x, y), Q(x, y)) Prof. Nicolas Dobigeon Complex variables - LT & ZT 5 / 96

6 Some Generalities Limits - continuity Outline Some Generalities Introduction Limits - continuity Usual functions Holomorphic functions Integration and Cauchy theorem Residue theorem Laplace transform Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 6 / 96

7 Some Generalities Limits - continuity Limits - continuity C is a vector space on R equipped with the norm z = z. Let f define a complex variable function and z 0 = x 0 + iy 0 and l two complex numbers. Definition: limit means: lim z z 0 f (z) = l or f (z) z z0 l ε > 0, η > 0, z z 0 < η = f (z) l < ε Definition: continuity f continue at z 0 lim z z 0 f (z) = f (z 0) P(x, y) and Q(x, y) continue at (x 0, y 0) Prof. Nicolas Dobigeon Complex variables - LT & ZT 7 / 96

8 Some Generalities Limits - continuity Limits - continuity Without any demonstration, we will admit that the standard operations on the limits or continuous functions are the same as those obtained for functions from R 2 R or from R R Warning! If P(x, y) is continuous at the point (x 0, y 0), then { x P(x, y0) is continuous at x = x 0 The reciprocal is wrong! y P(x 0, y) is continuous at y = y 0 Prof. Nicolas Dobigeon Complex variables - LT & ZT 8 / 96

9 Some Generalities Limits - continuity Complex infinity The complex infinity denoted is the unique complex number ensuring the following properties with a C: =, = /a =, a/ = 0, a = Representation on the Poincaré sphere, Extensions of the limit and neighboring definitions around infinity. Prof. Nicolas Dobigeon Complex variables - LT & ZT 9 / 96

10 Usual functions Algebraic functions Outline Some Generalities Usual functions Algebraic functions Functions defined by power series Multivalued functions (or multifunctions) Holomorphic functions Integration and Cauchy theorem Residue theorem Laplace transform Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 10 / 96

11 Usual functions Algebraic functions Algebraic functions Functions Définition Continuiy Associated T G z z + a C C Translation z a z C C Similarity z 1 z C C Inversion then symetry Ox z az+b cz+d C \ { } d c C \ { } d c... Prof. Nicolas Dobigeon Complex variables - LT & ZT 11 / 96

12 Usual functions Functions defined by power series Outline Some Generalities Usual functions Algebraic functions Functions defined by power series Multivalued functions (or multifunctions) Holomorphic functions Integration and Cauchy theorem Residue theorem Laplace transform Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 12 / 96

13 Usual functions Functions defined by power series Exponential function Definition e z = z n n! n=0 Properties e z z=x = e x e z 1+z 2 = e z 1 e z 2 e x+iy = e x (cos y + i sin y) e z = 1 e z We have the same functional relations as in R. Prof. Nicolas Dobigeon Complex variables - LT & ZT 13 / 96

14 Usual functions Functions defined by power series Hyperbolic and trigonometric functions Hyperbolic functions ch z = ez + e z, sh z = ez e z 2 2 Example: resolution of ch z = 0. Trigonometric functions cos z = eiz + e iz, sin z = eiz e iz 2 2i Example: resolution of sin z = 2., thz = shz chz, tan z = sin z cos z Prof. Nicolas Dobigeon Complex variables - LT & ZT 14 / 96

15 Usual functions Functions defined by power series Hyperbolic and trigonometric functions Properties Functions Definition set Continuity set exp C C ch C C sh C C th C\ { i ( π, k Z } C\ { i ( π, k Z } 2 2 cos C C sin C C tan C\ { π C\ { π + kπ, k Z} 2 2 Changing rules cos iz = ch z sin iz = i sh z tan iz = i th z and ch iz = cos z sh iz = i sin z th iz = i tan z Prof. Nicolas Dobigeon Complex variables - LT & ZT 15 / 96

16 Usual functions Multivalued functions (or multifunctions) Outline Some Generalities Usual functions Algebraic functions Functions defined by power series Multivalued functions (or multifunctions) Holomorphic functions Integration and Cauchy theorem Residue theorem Laplace transform Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 16 / 96

17 Usual functions Multivalued functions (or multifunctions) Multivalued function To any z of C, a unique value of e z corresponds. However, to any z of C, an infinity of values of arg z corresponds. To distinguish between these two cases, we are defining the so-called mono-valued vs. multivalues functions. Definitions A function f is named mono-valued if to any value z a unique value of f (z) corresponds. A function f is named multi-valued (aka multifunctions) if to any value z several distinct values of f (z) correspond. Prof. Nicolas Dobigeon Complex variables - LT & ZT 17 / 96

18 Usual functions Multivalued functions (or multifunctions) Multivalued functions Examples The argument function : C R z arg z is a multi-valued function. The functions introduce earlier are mono-valued. Study of the multivalued functions To study the multivalued functions, we make them mono-valued by defining its restrictions (or branches ) of rank k. Prof. Nicolas Dobigeon Complex variables - LT & ZT 18 / 96

19 Usual functions Multivalued functions (or multifunctions) Argument function The branch of rank k of the argument function is C\Ox + ]2kπ, 2 (k + 1) π[ z θ = arg k z Remarks The half-axe Ox + is called the branch cut. When k = 0, the restriction is called principal branch. Prof. Nicolas Dobigeon Complex variables - LT & ZT 19 / 96

20 Usual functions Multivalued functions (or multifunctions) Argument function: other definition (more general) The branch of rank k of the argument function is C\D α ]α + 2kπ, α + 2 (k + 1) π[ z θ = arg k,α z Remarks With this definition, the half-line (or ray) D α with origin O and angle α is the branch cut. Prof. Nicolas Dobigeon Complex variables - LT & ZT 20 / 96

21 Usual functions Multivalued functions (or multifunctions) Multifunctions Definitions Continuity values: values on the upper and lower sides of the branch cut. The point O at the origin of the branch cut is called the branch point. Remarks How to represent the branch cut? Closed paths enclosing the branch point branch change [WARNING] Closed paths enclosing the branch point no branch change Prof. Nicolas Dobigeon Complex variables - LT & ZT 21 / 96

22 Usual functions Multivalued functions (or multifunctions) Power functions The branch of rank k of z z 1 n is { C \ Ox + S k z z 1 n (k) = z 1 n e i 1 n arg k (z) = ρ 1 n e i θ n e i 2kπ n θ ]0, 2π[ This mapping is a bijective function from C \ Ox + in the open circular sector S k delimited by the two lines D 2kπ n and D 2(k+1)π n coming from O and making angles 2kπ n and 2(k+1)π n with Ox +. Prof. Nicolas Dobigeon Complex variables - LT & ZT 22 / 96

23 Usual functions Multivalued functions (or multifunctions) Power functions Extensions Function z (z a) 1 n. Function z (z a) α, α R. Example Restriction of z (z + 1) 1 2. Prof. Nicolas Dobigeon Complex variables - LT & ZT 23 / 96

24 Usual functions Multivalued functions (or multifunctions) Logarithm function The restriction of rank k of z log(z) is C \ Ox + B k z = z e iθ+i2kπ log k (z) = ln z + arg k (z) = ln ρ + iθ + i2kπ where B k is the open strip-like set defined by: {z / Imz ]2kπ, 2(k + 1)π[}. Extension Function z z α, α C defined by z α k = e α log k (z). Prof. Nicolas Dobigeon Complex variables - LT & ZT 24 / 96

25 Holomorphic functions Differentiable functions of two variables (reminders...) Outline Some Generalities Usual functions Holomorphic functions Differentiable functions of two variables (reminders...) Derivative of a complex variable function Holomorphic functions Complement : harmonic functions Integration and Cauchy theorem Residue theorem Laplace transform Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 25 / 96

26 Holomorphic functions Differentiable functions of two variables (reminders...) Differentiable functions of two variables A function P (x, y) is differentiable at the point (x 0, y 0 ) when it is defined in an open set containing this point and: P = A (x 0, y 0 ) h + B (x 0, y 0 ) k + (h, k) ε (h, k) with et P = P (x 0 + h, y 0 + k) P (x 0, y 0 ) lim ε (h, k) = 0 (h,k) 0 Prof. Nicolas Dobigeon Complex variables - LT & ZT 26 / 96

27 Holomorphic functions Derivative of a complex variable function Outline Some Generalities Usual functions Holomorphic functions Differentiable functions of two variables (reminders...) Derivative of a complex variable function Holomorphic functions Complement : harmonic functions Integration and Cauchy theorem Residue theorem Laplace transform Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 27 / 96

28 Holomorphic functions Derivative of a complex variable function Definition f (z) differentiable at z 0 if and only if Definition of the differentiability exists. It is denoted f (z) f (z 0) lim z z 0 z z 0 f (z 0) = lim z z0 f (z) f (z 0) z z 0 Example 1 f (z) = z z z 0 lim = 1, hence f est dérivable en z z z 0. 0 z z 0 Example 2 f (z) = z 2 z 2 z0 2 lim z z 0 z z 0 = lim z z0 (z + z 0) = 2z 0, hence f est dérivable en z 0. Prof. Nicolas Dobigeon Complex variables - LT & ZT 28 / 96

29 Holomorphic functions Derivative of a complex variable function Definition of the differentiability Counter example g(z) = z z z 0 z z 0 = (x x0) i (y y0) (x x 0) + i (y y 0) = 1 i y y 0 x x i y y 0 x x 0 which depends on the slope m of the path, thus f is not differentiable at z 0. z z 0 lim z z 0 z z 0 = 1 im 1 + im does not exist Prof. Nicolas Dobigeon Complex variables - LT & ZT 29 / 96

30 Holomorphic functions Derivative of a complex variable function Necessary and sufficient condition Property A complex variable function f is differentiable at the point z 0 = x 0 + iy 0 if and only if P (x, y) et Q (x, y) are differentiable at the point (x 0, y 0) and the Cauchy conditions are fulfilled: { P x P y (x0, y0) = Q y (x0, y0) = Q x (x0, y0) (x0, y0) Remark The demonstration of this condition allows ones to obtain f (z 0) = P Q (x0, y0) + i (x0, y0) x x f (z 0) = Q y P (x0, y0) i (x0, y0) y Prof. Nicolas Dobigeon Complex variables - LT & ZT 30 / 96

31 Holomorphic functions Holomorphic functions Outline Some Generalities Usual functions Holomorphic functions Differentiable functions of two variables (reminders...) Derivative of a complex variable function Holomorphic functions Complement : harmonic functions Integration and Cauchy theorem Residue theorem Laplace transform Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 31 / 96

32 Holomorphic functions Holomorphic functions Holomorphic functions Definition A complex variable function is said holomorphic on an open set A of C if it is differentiable in any point of A. Notation: f H/A. Properties The properties are the same as those related to differentiable in R. Let define f and g H/A. λf + µg H/A et (λf + µg) = λf + µg fg H/A et (fg) = f g + fg If z A, g (z) 0, then: 1 g H/A et ( ) 1 = g g g 2 If f H/A, g H/f (A), then: (g f ) H/A et (g f ) = (g f ) f If f is bijective from A onto f (A), then: ( f 1 H/f (A) et f 1) 1 = f f 1 Prof. Nicolas Dobigeon Complex variables - LT & ZT 32 / 96

33 Holomorphic functions Holomorphic functions Differentiability of usual functions Algebraic functions One formally differentiates with respect to z as for the real variable function with respect to x : (az) = a (z m ) = mz m 1, m Z Functions defined by series expansion Theorem of differentiability of power series: The function f (z) = a 0 + a 1z a nz n +... of convergence radius R is holomorphic on the open disk d (O, R). Its derivative is the sum of the term-wise differential series. Thus (e z ) = e z (chz) = shz (cos z) = sin z etc... One derives with respect to z as one derives in R with respect to x. Prof. Nicolas Dobigeon Complex variables - LT & ZT 33 / 96

34 Holomorphic functions Holomorphic functions Differentiability of multifunctions Derivative of log k z Z = log k (z) = ln ρ + iθ + 2ikπ defined from C \ Ox + to B k. One reminds that exp (log k (z)) = z. By the reciprocal formula, the derivative is given Thus: z = f (Z) = z = f (Z) Z = f 1 (z) = Z 1 = f (f 1 (z)) z = exp (Z) = z = exp (Z) Z = log k (z) = Z 1 = exp (log k (z)) = 1 z The additive constant disappears. Thus: log k z holomorphic on C \ Ox + et (log k ) (z) = 1 z Prof. Nicolas Dobigeon Complex variables - LT & ZT 34 / 96

35 Holomorphic functions Holomorphic functions Differentiability of multifunctions Derivative of z α (k), α C z α (k) = exp (α log k (z)) By differentiability of compound functions, one obtains: [ z α (k)] = [α [log k (z)]] exp [α log k (z)] Thus: [ z α (k)] = α z z α (k) The derivative owns the same multiplicative constant. Thus z α (k) holomorphic on C \ Ox + et [ z α (k)] = α z z α (k) Prof. Nicolas Dobigeon Complex variables - LT & ZT 35 / 96

36 Holomorphic functions Complement : harmonic functions Outline Some Generalities Usual functions Holomorphic functions Differentiable functions of two variables (reminders...) Derivative of a complex variable function Holomorphic functions Complement : harmonic functions Integration and Cauchy theorem Residue theorem Laplace transform Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 36 / 96

37 Holomorphic functions Complement : harmonic functions Complement : harmonic functions If we had time... Prof. Nicolas Dobigeon Complex variables - LT & ZT 37 / 96

38 Integration and Cauchy theorem Generalities Outline Some Generalities Usual functions Holomorphic functions Integration and Cauchy theorem Generalities Jordan lemmas Integral of holomorphic functions Residue theorem Laplace transform Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 38 / 96

39 Integration and Cauchy theorem Generalities Path A path of C is continuous function γ : [a, b] C, where [a, b] is an interval of R. If γ(a) = γ(b), γ is a closed path. γ is piecewise C 1 if γ (t) exists and is continuous on the intervals [t j 1, t j ] of R with t 0 = a < t 1 <... < t n = b. Prof. Nicolas Dobigeon Complex variables - LT & ZT 39 / 96

40 Integration and Cauchy theorem Generalities Complex curvilinear integral Let f (z) a function defined on a path γ which is piecewise-c 1 Let n k=1 z k 1z k define a subdivision of this path with ξ k z k 1 z k, z k = γ(t k ), z 0 = γ (a) and z n = γ (b). Definition: γ f (z)dz = lim with max k n k=1 n f (ξ k )(z k z k 1 ) z k z k 1 n 0 Prof. Nicolas Dobigeon Complex variables - LT & ZT 40 / 96

41 Integration and Cauchy theorem Generalities With the following notations Complex curvilinear integral z k = x k + iy k z k z k 1 = x k + i y k ξ k = a k + ib k f (ξ k ) = P(a k, b k ) + iq(a k, b k ) it yields γ n f (z)dz = lim P(a k, b k ) x k Q(a k, b k ) y k n k=1 n +i lim Q(a k, b k ) x k + P(a k, b k ) y k n + k=1 with max k x k 0 and max y k 0. Hence k f (z)dz = (Pdx Qdy) + i (Qdx + Pdy) γ γ γ Prof. Nicolas Dobigeon Complex variables - LT & ZT 41 / 96

42 Integration and Cauchy theorem Generalities Sufficient condition of existence In practice: γ is parametrized Usual paths Complex curvilinear integral P and Q continious on γ or f continious on γ γ f (z)dz = b a f (γ (t)) γ (t) dt Line segment parallel to the X-axis, z = x + iy 0, x [x 1, x 2] Line segment parallel to the Y-axis, z = x 0 + iy, y [y 1, y 2] Arc of radius R 0 z = R 0e iθ, θ [θ 1, θ 2] Line segment coming from the origin z = ρe iθ 0, ρ [ρ 1, ρ 2] Prof. Nicolas Dobigeon Complex variables - LT & ZT 42 / 96

43 Integration and Cauchy theorem Generalities Complex curvilinear integral Elementary properties of integrals a) Linearity (λf (z) + µg(z))dz = λ f (z)dz + µ g(z)dz γ γ γ b) Sense of the pathγ γ = γ + followed in the reverse sense. c) Integral of a constant f (z) = K f (z)dz = f (z)dz γ γ + n f (z k )(z k z k 1 ) = (z n z 0)K = (γ(b) γ(a))k k=1 Prof. Nicolas Dobigeon Complex variables - LT & ZT 43 / 96

44 Integration and Cauchy theorem Jordan lemmas Outline Some Generalities Usual functions Holomorphic functions Integration and Cauchy theorem Generalities Jordan lemmas Integral of holomorphic functions Residue theorem Laplace transform Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 44 / 96

45 Integration and Cauchy theorem Jordan lemmas Jordan lemmas 1st Lemma Jordan Assumptions C r (a, r) arc of center a and radius r lim r 0( resp. ) sup Cr (z a) f (z) = 0 Conclusion lim r 0( resp. ) C r f (z)dz = 0 Proof: f (z)dz = C r β α β α f (a + re iθ )rie iθ dθ rf (a + re iθ ) dθ (β α) sup C r (z a) f (z) Prof. Nicolas Dobigeon Complex variables - LT & ZT 45 / 96

46 Integration and Cauchy theorem Jordan lemmas Assumption lim sup Cr f (z) = 0 Jordan lemmas 2nd Jordan lemmas Conclusions Proof: lim C r e imz f (z)dz = 0 pour m > 0 et C r = C + r lim C r e imz f (z)dz = 0 pour m < 0 et C r = C r lim C r e mz f (z)dz = 0 pour m < 0 et C r = C d r lim C r e mz f (z)dz = 0 pour m > 0 et C r = C g r I r = π e imz f (z)dz = e imreiθ f (re iθ )ire iθ dθ C r 0 2r sup f (z) C r π 2 0 e mr sin θ dθ sup f (z) π C r m (1 e mr ) (car sin θ 2θ π ) Prof. Nicolas Dobigeon Complex variables - LT & ZT 46 / 96

47 Integration and Cauchy theorem Integral of holomorphic functions Outline Some Generalities Usual functions Holomorphic functions Integration and Cauchy theorem Generalities Jordan lemmas Integral of holomorphic functions Residue theorem Laplace transform Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 47 / 96

48 Integration and Cauchy theorem Integral of holomorphic functions Cauchy theorem 1-connected (or simply connected) domain Assumptions f holomorphic on Ω, non-null open space of C Let D Ω define a simply-connected domain of contour C Conclusion C f (z)dz = 0 Proof (by the use of the Green-Riemann formula) ( B Adx + Bdy = C + D x A ) dxdy y Prof. Nicolas Dobigeon Complex variables - LT & ZT 48 / 96

49 Integration and Cauchy theorem Integral of holomorphic functions Cauchy theorem n-connected domain - Generalization Example of a 2-connected domain f (z)dz = f (z)dz + C C + 1 C 2 f (z)dz = 0 Oriented contour τ tangent vector n oriented interior normal ( τ, n ) = + π 2 For δd = C + 1 C 2, it yields δd f (z)dz = 0 Prof. Nicolas Dobigeon Complex variables - LT & ZT 49 / 96

50 Integration and Cauchy theorem Integral of holomorphic functions Cauchy theorem Application Let f define a holomorphic function on a 1-connected domain D. a) Definition of b a f (z)dz Let a and b define two points of D. Let γ 1, γ 2 define two paths inside D with origin a and end point b. Then f (z)dz = f (z)dz = γ 1 γ 2 b a f (z)dz b) Definition of F z0 (u) = u z 0 f (z)dz, u C F z0 (u) is independent of the path from z 0 to u included in D F z0 (u) is a primitive of f (z) such that F z 0 (u) = f (u). Prof. Nicolas Dobigeon Complex variables - LT & ZT 50 / 96

51 Residue theorem Theorem for a bounded domain D Outline Some Generalities Usual functions Holomorphic functions Integration and Cauchy theorem Residue theorem Theorem for a bounded domain D Application to integral calculus Application to the sum of a series Laplace transform Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 51 / 96

52 Residue theorem Theorem for a bounded domain D Residue theorem Assumptions f holomorphic on Ω\ z j, Ω non-empty open set of C j z j isolated singularities of f D Ω 1-connected domain of contour D inside Ω Conclusion f (z)dz = 2iπ D resf (z + j ) with (definition of resf (z j )) : z j D 1 resf (z j ) = lim r 0 2iπ C + (z j,r) f (z)dz Prof. Nicolas Dobigeon Complex variables - LT & ZT 52 / 96

53 Residue theorem Theorem for a bounded domain D Remarks and definition Isolated singularities (IS, or isolated singular point) z j is an IS of f (z) if and only if r > 0 such that f is holomorphic in d(z j, r)\ {z j }, where d(z j, r) stands for the disc of center z i and radius r i. Computing the residue thanks to the Laurent series If z j is an IS, one admits that f has a Laurent series in d(z j, r)\{z j } : Thus, it comes: C + (z j,r) f (z) = f (z)dz = b n (z z n=1 j ) n + n=0 n=1 C + a n (z z j ) n b n (z z j ) n dz + n=0 C + a n (z z j ) n dz Prof. Nicolas Dobigeon Complex variables - LT & ZT 53 / 96

54 Residue theorem Theorem for a bounded domain D We set z z j = re iθ and it yields n=1 2π 0 Remarks and definition b n idθ r n 1 e + i i(n 1)θ n=0 2π All the integrals are null (straightforward...) except: Thus : 2π 0 C + (z j,r) b n idθ r n 1 e i(n 1)θ with n = 1 f (z)dz = 2π 0 0 a n r n+1 e i(n+1)θ dθ b 1 idθ = 2iπb 1 Conclusion : resf (z j ) is the coefficient of the term 1 z z j part of the Laurent series of f. of the main Prof. Nicolas Dobigeon Complex variables - LT & ZT 54 / 96

55 Residue theorem Theorem for a bounded domain D Remarks and definition Computing the residue in case of a pole of order p One computes the Taylor series of ϕ(z) = (z z j ) p f (z) which is holomorphic in V (z j ) ϕ(z) = ϕ(z j ) (z z j) p 1 ϕ (p 1) (p 1)! (z j ) +... As a consequence, the Laurent series of f is: thus f (z) = ϕ(z (p 1) j) (z z j ) p ϕ(z j ) (p 1)!(z z j ) +... resf (z j ) = 1 (p 1)! ϕ(p 1) (z j ) = 1 (p 1)! d p 1 dz [(z z p 1 j ) p f (z)] In practice: for p > 2, one compute the Laurent series, for p = 2, one can use resf (z j ) = d dz (z z j) 2 f (z) z=zj, for p = 1, one has resf (z j ) = lim z zj (z z j )f (z) z=zj Prof. Nicolas Dobigeon Complex variables - LT & ZT 55 / 96

56 Residue theorem Theorem for a bounded domain D Remarks and definition Interesting particular case : One expands Q(z) : z j pole of order 1, f (z) = P(z) Q(z), P(z j) 0 thus Q(z) = 0 + (z z j )Q (z j ) + (z z j) 2 Q (z j ) ! lim (z z j )f (z) = P(z j) z z j Q (z j ) This formula is interesting for some residue calculus, such as f (z) = 1 sin z en z = 0. Indeed: resf (0) = P(0) Q (0) = 1 cos 0 = 1 Prof. Nicolas Dobigeon Complex variables - LT & ZT 56 / 96

57 Residue theorem Application to integral calculus Outline Some Generalities Usual functions Holomorphic functions Integration and Cauchy theorem Residue theorem Theorem for a bounded domain D Application to integral calculus Application to the sum of a series Laplace transform Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 57 / 96

58 Residue theorem Application to integral calculus Integrals of the form: I = f (x)dx Very often, one defines f (z) and the contour which consists of a straight line associated with I and a circular parts which close path. Example: Computing I = + x x dx Prof. Nicolas Dobigeon Complex variables - LT & ZT 58 / 96

59 Residue theorem Application to integral calculus Integrals defined by a multifunction Example: show that, for a ]0, 1[ J = 0 x a x dx = π sin (πa) Prof. Nicolas Dobigeon Complex variables - LT & ZT 59 / 96

60 Residue theorem Application to integral calculus Trigonometric integrals I = 2π 0 R(cos θ, sin θ)dθ where R is a rational fraction. One sets z = e iθ and one derives cos θ and sin θ as functions of z. It consists of computing an integral on the unit circle. Example : show that J = 2π 0 dθ sin θ = π 2 Prof. Nicolas Dobigeon Complex variables - LT & ZT 60 / 96

61 Residue theorem Application to the sum of a series Outline Some Generalities Usual functions Holomorphic functions Integration and Cauchy theorem Residue theorem Theorem for a bounded domain D Application to integral calculus Application to the sum of a series Laplace transform Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 61 / 96

62 Residue theorem Application to the sum of a series Application to the sum of a series See exercise session. Prof. Nicolas Dobigeon Complex variables - LT & ZT 62 / 96

63 Laplace transform Definition Outline Some Generalities Usual functions Holomorphic functions Integration and Cauchy theorem Residue theorem Laplace transform Definition Properties Inverse Laplace transform Applications Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 63 / 96

64 Laplace transform Definition Definition Set of the (Laplace) transformable functions E is the set of the functions f defined on R + such that f is locally integrable, i.e., A f (t)dt <, A 0 It exists x 0 such that e x0t f (t)dt < 0 Laplace transform For f E, one defines its Laplace transform as Notation: F (p) = TL(f (t)) F (p) 0 e pt f (t)dt p C Prof. Nicolas Dobigeon Complex variables - LT & ZT 64 / 96

65 Laplace transform Definition Definition Convergences (simple) Convergence Thereom 1 If F (p) exists for p = p 0 = x 0 + iy 0 then F (p) exists p such that Rep > Rep 0 = x 0 Consequence : {x R, F (p) < } admits a lower bound denoted x c and called abscissa of (simple) convergence of F. Absolute convergence Theorem 2 If 0 e pt f (t) dt exists for p = p0 = x 0 + iy 0 then 0 e pt f (t) dt exists p such that Rep > Rep0 = x 0 Consequence : { x R, e pt f (t) dt < } admits a lower bound denoted 0 x ca and called abscissa of absolute convergence of F (obviously, x c x ca) Example: f (t) = e kt sin [ e kt], k > 0, x c = 0 and x ca = k. Remark: one often has x c = x ca. Prof. Nicolas Dobigeon Complex variables - LT & ZT 65 / 96

66 Laplace transform Definition Definition Fundamental theorem If f (t) is piecewise continuous on R +, then F (p) = e pt f (t)dt is holomorphic on ]x 0 c, + [ and then it is infinitely differentiable on ]x c, + [ with d n F (p) dp = d n n 0 dp [e pt f (t)] dt n Consequence: deriving x c from F (p) If F (p) a function of the complex variable p is the Laplace transform of a function f (t) which admits isolated singularities s k and branching points r j in C, then x c = sup Re(s k, r j ) Examples: F (p) = 1 p(p 2) x c = 2 F (p) = 1 p+1 x c = 0 Prof. Nicolas Dobigeon Complex variables - LT & ZT 66 / 96

67 Laplace transform Properties Outline Some Generalities Usual functions Holomorphic functions Integration and Cauchy theorem Residue theorem Laplace transform Definition Properties Inverse Laplace transform Applications Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 67 / 96

68 Laplace transform Properties a) Linearity Usual properties TL (λf + µg) =λf (p) + µg(p) Generally, abscissa of convergence x c = sup(x cf, x cg ). b) Derivation * with respect to p TL {( 1) n t n f (t)} = d n dp n F (p) * with respect to t (f continuous on [0, + [) TL [f (t)] = pf (p) f (0 + ) Generalization: [ ] TL f (n) (t) = p n F (p) p n 1 f (0 + )... f (n 1) (0 + ) Application: resolution of linear differential equations Prof. Nicolas Dobigeon Complex variables - LT & ZT 68 / 96

69 Laplace transform Properties Usual properties c) Integration * LT of a primitive [ t ] TL f (u)du = F (p) 0 p Abscissa of convergence: sup(x c, 0) * Primitive of a LT [ ] f (t) TL = t p F (u)du Prof. Nicolas Dobigeon Complex variables - LT & ZT 69 / 96

70 Laplace transform Properties Usual properties d) Translation * with respect to p Abscissa of convergence: x c + Re(a) * with respect to t TL [ e at f (t) ] = F (p a) TL [f (t a)u(t a)] = e ap F (p) Abscissa of convergence: x c Remark: Application to periodic functions e) Scaling [ ( t )] TL f = kf (kp) k > 0 k Abscissa of convergence: x c k Prof. Nicolas Dobigeon Complex variables - LT & ZT 70 / 96

71 Laplace transform Properties f) Convolution Usual properties [ t ] TL f (u)g(t u)du = F (p)g(p) 0 g) Theorems of the initial and final values lim f (t) = lim pf (p) t 0 + p lim f (t) = lim pf (p) t p 0 h) Transform of series t Series of general term a n n n! with abscissa of convergence R c = [ ] TL n=1 a n tn n! = n=1 Example: show that TL [ ] sin ωt t = Arctg ω p Use two methods: series expansion and TL a n p n+1 [ ] x(t) t Prof. Nicolas Dobigeon Complex variables - LT & ZT 71 / 96

72 Laplace transform Properties Some Laplace transforms Function TL Convergence 1 U(t) p x c = 0 e αt 1 p α x c = Reα e iωt 1 p iω x c = 0 p ch (αt) p 2 α x 2 c = sup Re(α, α) α sh (αt) p 2 α x 2 c = sup Re(α, α) p cos ωt p 2 +ω x 2 c = 0 ω sin ωt p 2 +ω x 2 c = 0 1 t p x c = 0 t n n!, n N p n+1 x c = 0 t α, α R Γ(α+1) p α+1 with Γ(x) = 0 e t t x 1 dt et Γ(n + 1) = nγ(n) = n! Prof. Nicolas Dobigeon Complex variables - LT & ZT 72 / 96

73 Laplace transform Inverse Laplace transform Outline Some Generalities Usual functions Holomorphic functions Integration and Cauchy theorem Residue theorem Laplace transform Definition Properties Inverse Laplace transform Applications Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 73 / 96

74 Laplace transform Inverse Laplace transform Inversion formula X (p) = x(t)e pt dt = x(t)e at e j2πft dt 0 0 with p = a + j2πf Analogy with the Fourier transform Hence : and thus the inversion formula: X (f ) = TF (x(t)) = R x(t)e j2πft x(t) = TF 1 (X (f )) = R X (f )e+j2πft df X (p) = TF [ x(t)e at U(t) ] x(t)u(t) = 1 2iπ D X (p)ept dp One applies the residue theorem with X (p)e pt. Example: X (p) = 1 p Prof. Nicolas Dobigeon Complex variables - LT & ZT 74 / 96

75 Laplace transform Applications Outline Some Generalities Usual functions Holomorphic functions Integration and Cauchy theorem Residue theorem Laplace transform Definition Properties Inverse Laplace transform Applications Z transform Prof. Nicolas Dobigeon Complex variables - LT & ZT 75 / 96

76 Laplace transform Applications Initial conditions Laplace transform Differential equations with constant coefficients y (n) + a 1 y (n 1) a n y(t) = f (t) y(0) = b 0, y (0) = b 1,..., y (n 1) (0) = b (n 1) TL [a n y(t)] = a n Y (p) TL [ y (n) (t) ] = p n Y (p) p n 1 y(0 + )... y (n 1) (0 + ) TL [Ω n (y)] = Ω n (p)y (p) Q n 1 (p) TL [f (t)] = F (p) Algebraic problem Ω n (p)y (p) = Q n 1 (p) + F (p) Y (p) = Qn 1(p) Ω n(p) + F (p) Ω n(p) = Y 1(p) + Y 2 (p) Prof. Nicolas Dobigeon Complex variables - LT & ZT 76 / 96

77 Laplace transform Applications Differential equations with constant coefficients a) Y 1 (p) Algebraic fraction Y 1 (p) = Q n 1 (p) r i=1 (p p i) k i where p i is a k i -order root with r i=1 k i = n Partial fraction decomposition : where Y 1 (p) = { r i=1 A i1 p p i + A i2 (p p i ) A iki (p p i ) k i y 1 (t) = r i=1 ep i t [ A i1 + A i2 t A iki t k i 1 ] b) Y 2 (p) = F (p) Ω n(p) = F (p) 1 Ω n(p) thus: y 2 (t) = t 0 f (u)r n (t u)du Hence, the solution of the problem is y(t) = y 1 (t) + y 2 (t) Prof. Nicolas Dobigeon Complex variables - LT & ZT 77 / 96 }

78 Laplace transform Applications Partial differential equation of several variables The LT allows one to reduce the equation with respect to one dimension. Example: Two-dimensional spatio-temporal problem: string vibration f (x, t) Initial conditions Conditions at limits 2 f x 1 2 f 2 c 2 t = 0 2 f (x, 0) = ϕ(x) f (x, 0) = ψ(x) t f (, t) = 0 f (0, t) = g(t) Prof. Nicolas Dobigeon Complex variables - LT & ZT 78 / 96

79 Laplace transform Applications Partial differential equation of several variables Solution thanks to the LT (p is a considered as a parameter) F (x, p) = e pt f (x, t)dt 0 [ ] f TL t = pf (x, p) f (x, 0) = pf (x, p) ϕ(x) [ 2 ] f TL (x, t) t2 [ 2 ] f TL (x, t) x 2 = = p 2 F (x, p) pf (x, 0) f (x, 0) t = p 2 F (x, p) pϕ(x) ψ(x) 0 = 2 x 2 e pt 2 f (x, t) x 2 dt 0 e pt f (x, t)dt = d 2 F (x, p) dx 2 Prof. Nicolas Dobigeon Complex variables - LT & ZT 79 / 96

80 Laplace transform Applications Partial differential equation of several variables One obtains with d 2 F (x,p) dx 2 p 2 F (x, p) = pϕ(x) + ψ(x) F (, p) = TL [f (, t)] = 0 G(p) = TL [g(t)] = TL[f (0, t)] = F (0, p) One-dimensional problem (differential equations + conditions at limits). Prof. Nicolas Dobigeon Complex variables - LT & ZT 80 / 96

81 Z transform Definition Outline Some Generalities Usual functions Holomorphic functions Integration and Cauchy theorem Residue theorem Laplace transform Z transform Definition Properties Inverse Z transform Applications Laplace and Z transforms Prof. Nicolas Dobigeon Complex variables - LT & ZT 81 / 96

82 Z transform Definition Definition Definition One defines the Z transform of a series x(n), n Z as: X (z) = + n= x(n)z n z C Notation: Remark: bilateral and unilateral TZ. X (z) = TZ(x(n)) Prof. Nicolas Dobigeon Complex variables - LT & ZT 82 / 96

83 Z transform Definition Definition Domain of convergence The domain of convergence is the set of complex numbers z such that the series X (z) converges. Reminder: Cauchy criterion + n lim un < 1 = u n converge n + n=0 One has a sufficient condition of convergence. Thanks to this criterion, one shows that the series X (z) converges once: 0 R x < z < R + x + Example: X (z) = + n=0 z n converges for z > 1 Prof. Nicolas Dobigeon Complex variables - LT & ZT 83 / 96

84 Z transform Properties Outline Some Generalities Usual functions Holomorphic functions Integration and Cauchy theorem Residue theorem Laplace transform Z transform Definition Properties Inverse Z transform Applications Laplace and Z transforms Prof. Nicolas Dobigeon Complex variables - LT & ZT 84 / 96

85 Z transform Properties Usual properties Linearity TZ(ax(n) + by(n)) = ax (z) + by (z) Convergence: if R + = min(r + x, R + y ) and R = max(r x, R y ), then the convergence domain contains ]R, R + [. Shifting TZ(x(n n 0 )) = z n0 X (z) Same domain of convergence as X (z). Scaling ( z ) TZ(a n x(n)) = X a Domain of convergence: a R x < z < a R + x Prof. Nicolas Dobigeon Complex variables - LT & ZT 85 / 96

86 Z transform Properties Usual properties Differentiability The Z transform defines a Laurent series which is infinitely differentiable term-by-term in its domain of convergence. Thus dx (z) TZ(nx(n)) = z dz Same domain of convergence as X (z). Convolution product The convolution between the series x(n) and y(n) is defined as: u(n) = x(n) y(n) = + k= x(k)y(n k) Thus TZ(x(n) y(n)) = X (z)y (z) The domain of convergence of U(z) can be larger than the intersection of domains of convergence of X (z) and Y (z), respectively. Prof. Nicolas Dobigeon Complex variables - LT & ZT 86 / 96

87 Z transform Inverse Z transform Outline Some Generalities Usual functions Holomorphic functions Integration and Cauchy theorem Residue theorem Laplace transform Z transform Definition Properties Inverse Z transform Applications Laplace and Z transforms Prof. Nicolas Dobigeon Complex variables - LT & ZT 87 / 96

88 Z transform Inverse Z transform Inverse Z transform The inverse Z transform is given by: x(n) = 1 X (z)z n 1 dz j2π C + where C is a closed path included into the domain of convergence Prof. Nicolas Dobigeon Complex variables - LT & ZT 88 / 96

89 Z transform Inverse Z transform TZ inverse Proof One has to compute the integrals J(n, k) = z n k 1 dz C + Thanks to the residue theorem, one shows that: { 0 si n k J(n, k) = j2π si n = k Hence: 1 j2π Remark : tables C + X (z)z n 1 dz = = ( ) 1 x(k)z k z n 1 dz j2π C + k= 1 x(k)j(n, k) j2π = x(n) k= Prof. Nicolas Dobigeon Complex variables - LT & ZT 89 / 96

90 Z transform Applications Outline Some Generalities Usual functions Holomorphic functions Integration and Cauchy theorem Residue theorem Laplace transform Z transform Definition Properties Inverse Z transform Applications Laplace and Z transforms Prof. Nicolas Dobigeon Complex variables - LT & ZT 90 / 96

91 Z transform Applications Discrete signal filtering See exercise session and/or later. Prof. Nicolas Dobigeon Complex variables - LT & ZT 91 / 96

92 Z transform Applications Recurrence relations Example: 1-st order system y(n) ay(n 1) = x(n) a < 1 The input of the system is chosen as: x(n) = b n U(n) with b < 1 where U(n) is the Heaviside step function. Compute y(n) for n 0 given that y(n) = 0 for n < 0. Determine the impulse response of the system h(n) such that y(n) = x(n) h(n). Prof. Nicolas Dobigeon Complex variables - LT & ZT 92 / 96

93 Z transform Laplace and Z transforms Outline Some Generalities Usual functions Holomorphic functions Integration and Cauchy theorem Residue theorem Laplace transform Z transform Definition Properties Inverse Z transform Applications Laplace and Z transforms Prof. Nicolas Dobigeon Complex variables - LT & ZT 93 / 96

94 Z transform Laplace and Z transforms Laplace and Z transforms Let x(t) define a causal signal whose Laplace transform is: X (p) = 0 x(t)e pt dt One samples this signal with period T and one denotes X (z) its Z transform: X (z) = x(nt )z n Then n=0 X (z) = res X (p) 1 e pt z 1 Prof. Nicolas Dobigeon Complex variables - LT & ZT 94 / 96

95 Z transform Laplace and Z transforms Laplace and Z transforms The formula of inverse Laplace transform provides x(t)u(t) = 1 X (p)e pt dp 2iπ D hence X (z) = = [ ] 1 x(nt )z n = X (p)e pnt dp z n 2iπ D n=0 1 2iπ D n=0 ( X (p) z 1 e pt ) n dp n=0 Once z 1 e pt < 1, on a X (z) = 1 2iπ D 1 X (p) 1 z 1 e pt dp = X (p) res 1 e pt z 1 Prof. Nicolas Dobigeon Complex variables - LT & ZT 95 / 96

96 Z transform Laplace and Z transforms Complex Variables Laplace Transform Z Transform Prof. Nicolas Dobigeon University of Toulouse IRIT/INP-ENSEEIHT nicolas.dobigeon@enseeiht.fr Prof. Nicolas Dobigeon Complex variables - LT & ZT 96 / 96