complex variables - laplace transform z transform

Complex VariablesLaplace Transform – Z Transform

Prof. Nicolas Dobigeon

University of ToulouseIRIT/INP-ENSEEIHT

http://www.enseeiht.fr/[email protected]

Prof. Nicolas Dobigeon Complex variables - LT & ZT 1 / 105

http://www.enseeiht.fr/~dobigeon

Framework

Organization

I 7 course & exercice sessions (1h45),

I 1 written exam (shared with Vector Analysis)

Bibliography

I Handout ENSEEIHT

I Books

I S. D. Chatterji, Cours d’analyse (vol. 2), Presses Polytechniques etUniversitaires Romandes, 1997

I Spiegel, Variables complexes (cours et problemes), Serie Schaum,McGraw Hill., 1973


Motivation

Applications

I Analysis and numerical calculus,

I Laplace transformCircuit theory,

I Z transformSampled systems,Digital filtering,Digital signal processing,

Prerequisites

I Usual algebra of complex numbers: properties, geometry associated withthe vector representation,

I Differentiable functions of two real variables,

I curvilinear integrals


Outline

Some Generalities

Usual functions

Holomorphic functions

Integration and Cauchy theorem

Residue theorem

Laplace transform

Z transform


Some Generalities

Outline

Some GeneralitiesIntroductionLimits - continuity

Usual functions



Residue theorem

Laplace transform

Z transform


Some Generalities

Introduction

Outline


Usual functions



Residue theorem

Laplace transform

Z transform


Some Generalities

Introduction

Complex plane (z-plane)

Complex plan is the plane equipped with the direct orthonormal basis(O; u, v). La correspondence

R2 → C(x , y) 7→ z = x + iy

is bijective.By a slight abuse of notation, the point M (x , y) and its affix z = x + iycoincide.If z 6= 0, the representation of the complex number z under the formmodulus/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.


Some Generalities

Introduction

Complex function of the z-variable

For any function f of complex variable

f :

C → C

z = x + iy 7→ f (z) = P(x , y) + iQ(x , y)

we can define a function F :

F :

R2 → R2

(x , y) 7→ F (x , y) = (P(x , y),Q(x , y))


Some Generalities

Limits - continuity

Outline


Usual functions



Residue theorem

Laplace transform

Z transform


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 z0 = x0 + iy0 and l twocomplex numbers.

Definition: limit

limz−→z0

f (z) = l or f (z) −→z−→z0

l

means:∀ε > 0, ∃η > 0, |z − z0| < η =⇒ |f (z)− l | < ε

Definition: continuity

f continue at z0 ⇐⇒ limz−→z0

f (z) = f (z0)

⇐⇒ P(x , y) and Q(x , y) continue at (x0, y0)


Some Generalities

Limits - continuity

Limits - continuity

Without any demonstration, we will admit that the standard operationson the limits or continuous functions are the same as those obtained forfunctions from R2 → R or from R→ R

Warning !If P(x , y) is continuous at the point (x0, y0), then

x 7→ P(x , y0) is continuous at x = x0

y 7→ P(x0, y) is continuous at y = y0

The reciprocal is wrong!


Some Generalities

Limits - continuity

Complex infinity

The complex infinity denoted ∞ is the unique complex number ensuringthe following properties with a ∈ C:

∞×∞ = ∞, |∞| =∞∞/a = ∞, a/∞ = 0, a×∞ =∞

I Representation on the Poincare sphere,

I Extensions of the limit and neighboring definitions around infinity.


Usual functions

Outline

Some Generalities

Usual functionsAlgebraic functionsFunctions defined by power seriesMultivalued functions (or multifunctions)



Residue theorem

Laplace transform

Z transform


Usual functions

Algebraic functions

Outline

Some Generalities




Residue theorem

Laplace transform

Z transform


Usual functions

Algebraic functions

Algebraic functions

Functions Definition Continuiy Associated TG

z 7−→ z + a C C Translationz 7−→ a z C C Similarityz 7−→ 1

z C∗ C∗ Inversion then symetry Ox

z 7−→ az+bcz+d C \

− d

c

C \

− d

c

...


Usual functions

Functions defined by power series

Outline

Some Generalities




Residue theorem

Laplace transform

Z transform


Usual functions


Exponential function

Definition

ez =∞∑n=0

zn

n!

Properties

ez |z=x = ex

ez1+z2 = ez1ez2

ex+iy = ex (cos y + i sin y)

e−z =1

ez

We have the same functional relations as in R.


Usual functions


Hyperbolic and trigonometric functions

Hyperbolic functions

ch z =ez + e−z

2, sh z =

ez − e−z

2, thz =

shz

chzExample: resolution of ch z = 0.

Trigonometric functions

cos z =e iz + e−iz

2, sin z =

e iz − e−iz

2i, tan z =

sin z

cos zExample: resolution of sin z = 2.


Usual functions


Hyperbolic and trigonometric functions

PropertiesFunctions Definition set Continuity set

exp C Cch C Csh C Cth C\

i(π2

+ kπ), k ∈ Z

C\i(π2

+ kπ), k ∈ Z

cos C Csin C Ctan C\

π2

+ kπ, k ∈ Z

C\π2

+ kπ, k ∈ Z

Changing rulescos iz = ch zsin iz = i sh z

tan iz = i th zand

ch iz = cos zsh iz = i sin zth iz = i tan z


Usual functions

Multivalued functions (or multifunctions)

Outline

Some Generalities




Residue theorem

Laplace transform

Z transform


Usual functions


Multivalued function

To any z of C, a unique value of ez corresponds. However, to any z ofC∗, an infinity of values of arg z corresponds. To distinguish betweenthese two cases, we are defining the so-called mono-valued vs.multivalues functions.

Definitions

I A function f is named mono-valued if to any value z a unique value off (z) corresponds.

I A function f is named multi-valued (aka multifunctions) if to any value zseveral distinct values of f (z) correspond.


Usual functions


Multivalued functions

Examples

I The argument function :

C∗ −→ Rz 7−→ arg z

is a multi-valued function.

I The functions introduce earlier are mono-valued.

Study of the multivalued functionsTo study the multivalued functions, we make them mono-valued by defining itsrestrictions (or “branches”) of rank k.


Usual functions


Argument function

The branch of rank k of the argument function is

C\Ox+ −→ ]2kπ, 2 (k + 1)π[

z 7−→ θ = argk z

Remarks

I The half-axe Ox+ is called the branch cut.

I When k = 0, the restriction is called “principal branch”.


Usual functions


Argument function: other definition (more general)

The branch of rank k of the argument function is

C\Dα −→ ]α + 2kπ, α + 2 (k + 1)π[

z 7−→ θ = argk,α z

Remarks

I With this definition, the half-line (or ray) Dα with origin O and angle α isthe branch cut.


Usual functions


Multifunctions

Definitions

I Continuity values: values on the upper and lower sides of the branch cut.

I The point O at the origin of the branch cut is called the branch point.

Remarks

I How to represent the branch cut?

I Closed paths enclosing the branch point → branch change [WARNING]

I Closed paths enclosing the branch point → no branch change


Usual functions


Power functions

The branch of rank k of z 7→ z1n is

C \ Ox+ → Sk

z 7→ z1n

(k) = |z |1n e i

1n argk (z) = ρ

1n e i

θn e i

2kπn

θ ∈ ]0, 2π[

This mapping is a bijective function from C \ Ox+ in the open circularsector Sk delimited by the two lines D 2kπ

nand D 2(k+1)π

ncoming from O

and making angles 2kπn and 2(k+1)π

n with Ox+.


Usual functions


Power functions

Extensions

I Function z 7→ (z − a)1n .

I Function z 7→ (z − a)α, α ∈ R.

Example

Restriction of z 7→ (z + 1)12 .


Usual functions


Logarithm function

The restriction of rank k of z 7→ log(z) is C \ Ox+ → Bk

z = |z |e iθ+i2kπ 7→ logk(z) = ln |z |+ argk(z)= ln ρ+ iθ + i2kπ

where Bk is the open strip-like set defined by:z / Imz ∈ ]2kπ, 2(k + 1)π[ .

Extension

I Function z 7→ zα, α ∈ C defined by zαk = eα logk (z).



Outline

Some Generalities

Usual functions

Holomorphic functionsDifferentiable functions of two variables (reminders...)Derivative of a complex variable functionHolomorphic functionsComplement : harmonic functions


Residue theorem

Laplace transform

Z transform



Differentiable functions of two variables (reminders...)

Outline

Some Generalities

Usual functions



Residue theorem

Laplace transform

Z transform



Differentiable functions of two variables (reminders...)

Differentiable functions of two variables

A function P (x , y) is differentiable at the point (x0, y0) when it isdefined in an open set containing this point and:

∆P = A (x0, y0) h + B (x0, y0) k + ‖(h, k)‖ ε (h, k)

with∆P = P (x0 + h, y0 + k)− P (x0, y0)

etlim

‖(h,k)‖→0ε (h, k) = 0



Derivative of a complex variable function

Outline

Some Generalities

Usual functions



Residue theorem

Laplace transform

Z transform




Definition of the differentiability

Definitionf (z) differentiable at z0 if and only if

limz→z0

f (z)− f (z0)

z − z0

exists. It is denoted

f ′ (z0) = limz→z0

f (z)− f (z0)

z − z0

Example 1f (z) = z

limz→z0

z − z0

z − z0= 1, hence f est derivable en z0.

Example 2f (z) = z2

limz→z0

z2 − z20

z − z0= lim

z→z0

(z + z0) = 2z0, hence f est derivable en z0.




Definition of the differentiability

Counter exampleg(z) = z

z − z0

z − z0=

(x − x0)− i (y − y0)

(x − x0) + i (y − y0)

=1− i y−y0

x−x0

1 + i y−y0x−x0

=1− im

1 + im

which depends on the slope m of the path, thus

limz→z0

z − z0

z − z0does not exist

⇒ f is not differentiable at z0.




Necessary and sufficient condition

PropertyA complex variable function f is differentiable at the point z0 = x0 + iy0 if andonly if

I P (x , y) et Q (x , y) are differentiable at the point (x0, y0) and

I the Cauchy conditions are fulfilled:∂P∂x

(x0, y0) = ∂Q∂y

(x0, y0)∂P∂y

(x0, y0) = − ∂Q∂x

(x0, y0)

RemarkThe demonstration of this condition allows ones to obtain

f ′ (z0) =∂P

∂x(x0, y0) + i

∂Q

∂x(x0, y0)

f ′ (z0) =∂Q

∂y(x0, y0)− i

∂P

∂y(x0, y0)




Outline

Some Generalities

Usual functions



Residue theorem

Laplace transform

Z transform





DefinitionA complex variable function is said holomorphic on an open set A of C if it isdifferentiable in any point of A. Notation: f ∈ H/A.

PropertiesThe properties are the same as those related to differentiable in R. Let define fand g ∈ H/A.I λf + µg ∈ H/A et (λf + µg)′ = λf ′ + µg ′

I fg ∈ H/A et (fg)′ = f ′g + fg ′

I If ∀z ∈ A, g (z) 6= 0, then:

1

g∈ H/A et

(1

g

)′= − g ′

g 2

I If f ∈ H/A, g ∈ H/f (A), then: (g f ) ∈ H/A et (g f )′ = (g ′ f ) f ′

I If f is bijective from A onto f (A), then:

f −1 ∈ H/f (A) et(f −1

)′=

1

f ′ f −1




Differentiability of usual functions

Algebraic functionsOne formally differentiates with respect to z as for the real variable functionwith respect to x :

(az)′ = a

(zm)′ = mzm−1, m ∈ Z

Functions defined by series expansionTheorem of differentiability of power series:The function f (z) = a0 + a1z + ...+ anz

n + ... of convergence radius R isholomorphic on the open disk d (O,R). Its derivative is the sum of theterm-wise differential series. Thus

(ez)′ = ez

(chz)′ = shz

(cos z)′ = − sin z

etc ...

One derives with respect to z as one derives in R with respect to x .




Differentiability of multifunctions

I Derivative of logk z

Z = logk (z) = ln ρ+ iθ + 2ikπ

defined from C \ Ox+ to Bk .

One reminds that exp (logk (z)) = z . By the reciprocal formula, thederivative is given

z = f (Z) =⇒ z ′ = f ′ (Z)

Z = f −1 (z) =⇒ Z ′ =1

f ′ (f −1 (z))

Thus:

z = exp (Z) =⇒ z ′ = exp (Z)

Z = logk (z) =⇒ Z ′ =1

exp (logk (z))=

1

z

The additive constant disappears. Thus:

logk z holomorphic on C \ Ox+ et (logk)′ (z) = 1z




Differentiability of multifunctions

I Derivative of zα(k), α ∈ C

zα(k) = exp (α logk (z))

By differentiability of compound functions, one obtains:[zα(k)

]′= [α [logk (z)]]′ exp [α logk (z)]

Thus: [zα(k)

]′=α

zzα(k)

The derivative owns the same multiplicative constant. Thus

zα(k) holomorphic on C \ Ox+ et[zα(k)

]′=α

zzα(k)



Complement : harmonic functions

Outline

Some Generalities

Usual functions



Residue theorem

Laplace transform

Z transform





If we had time...



Outline

Some Generalities

Usual functions


Integration and Cauchy theoremGeneralitiesJordan lemmasIntegral of holomorphic functions

Residue theorem

Laplace transform

Z transform



Generalities

Outline

Some Generalities

Usual functions



Residue theorem

Laplace transform

Z transform



Generalities

Path

I A path of C is continuous function γ : [a, b]→ C, where [a, b] is aninterval of R.

I If γ(a) = γ(b), γ is a closed path.

I γ is piecewise C 1 if γ′(t) exists and is continuous on the intervals[tj−1, tj ] of R with t0 = a < t1 < ... < tn = b.



Generalities

Complex curvilinear integral

Let f (z) a function defined on a path γ which is piecewise-C 1

Letn∪

k=1zk−1zk define a subdivision of this path with ξk ∈ zk−1zk ,

zk = γ(tk), z0 = γ (a) and zn = γ (b).

Definition: ∫γf (z)dz = lim

n→∞

n∑k=1

f (ξk)(zk − zk−1)

with maxk|zk − zk−1| →

n→∞0



Generalities


With the following notations

zk = xk + iyk

zk − zk−1 = ∆xk + i∆yk

ξk = ak + ibk

f (ξk) = P(ak , bk) + iQ(ak , bk)

it yields ∫γf (z)dz = lim

n→∞

n∑k=1

P(ak , bk)∆xk − Q(ak , bk)∆yk

+i limn→+∞

n∑k=1

Q(ak , bk)∆xk + P(ak , bk)∆yk

with maxk|∆xk | → 0 and max

k|∆yk | → 0. Hence∫

γ

f (z)dz =

∫γ

(Pdx − Qdy) + i

∫γ

(Qdx + Pdy)



Generalities


Sufficient condition of existence

P and Q continious on γor f continious on γ

In practice: γ is parametrized∫γf (z)dz =

∫ b

af (γ (t)) γ′ (t) dt

Usual paths

I Line segment parallel to the X-axis,z = x + iy0, x ∈ [x1, x2]

I Line segment parallel to the Y-axis,z = x0 + iy , y ∈ [y1, y2]

I Arc of radius R0

z = R0eiθ, θ ∈ [θ1, θ2]

I Line segment coming from the originz = ρe iθ0 , ρ ∈ [ρ1, ρ2]



Generalities


Elementary properties of integralsa) Linearity ∫

γ

(λf (z) + µg(z))dz = λ

∫γ

f (z)dz + µ

∫γ

g(z)dz

b) Sense of the pathγ ∫γ−

f (z)dz = −∫γ+

f (z)dz

γ− = γ+ followed in the reverse sense.

c) Integral of a constant f (z) = K

n∑k=1

f (zk)(zk − zk−1) = (zn − z0)K = (γ(b)− γ(a))K



Jordan lemmas

Outline

Some Generalities

Usual functions



Residue theorem

Laplace transform

Z transform



Jordan lemmas

Jordan lemmas1st Lemma Jordan

AssumptionsCr (a, r) arc of center a and radius rlimr→0( resp. ∞) supCr

|(z − a) f (z)| = 0

Conclusion

limr→0( resp. ∞)

∫Crf (z)dz = 0

Proof: ∣∣∣∣∫Cr

f (z)dz

∣∣∣∣ =

∣∣∣∣∣∫ β

α

f (a + re iθ)rie iθdθ

∣∣∣∣∣6

∫ β

α

∣∣rf (a + re iθ)∣∣ dθ

6 (β − α) supCr

|(z − a) f (z)|



Jordan lemmas

Jordan lemmas2nd Jordan lemmas

Assumptionlim∞ supCr

|f (z)| = 0

Conclusions

lim∞∫Cre imz f (z)dz = 0 pour m > 0 et Cr = C+

r

lim∞∫Cre imz f (z)dz = 0 pour m < 0 et Cr = C−r

lim∞∫Cremz f (z)dz = 0 pour m < 0 et Cr = C d

r

lim∞∫Cremz f (z)dz = 0 pour m > 0 et Cr = C g

r

Proof:

|Ir | =

∣∣∣∣∫Cr

e imz f (z)dz

∣∣∣∣ =

∣∣∣∣∫ π

0

e imre iθ f (re iθ)ire iθdθ

∣∣∣∣≤ 2r sup

Cr

|f (z)|∫ π

2

0

e−mr sin θdθ

≤ supCr

|f (z)| πm

(1− e−mr ) (car sin θ >2θ

π)



Integral of holomorphic functions

Outline

Some Generalities

Usual functions



Residue theorem

Laplace transform

Z transform




Cauchy theorem1-connected (or simply connected) domain

Assumptionsf holomorphic on Ω, non-null open space of CLet D ⊂ Ω define a simply-connected domain of contour C

Conclusion ∫Cf (z)dz = 0

Proof (by the use of the Green-Riemann formula)∫C+

Adx + Bdy =

∫ ∫D

(∂B

∂x− ∂A

∂y

)dxdy




Cauchy theoremn-connected domain - Generalization

Example of a 2-connected domain∫C

f (z)dz =

∫C+

1

f (z)dz +

∫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




Cauchy theoremApplication

Let f define a holomorphic function on a 1-connected domain D.

a) Definition of∫ b

af (z)dz

Let a and b define two points of D.Let γ1, γ2 define two paths inside D with origin a and end point b. Then∫

γ1

f (z)dz =

∫γ2

f (z)dz =

∫ b

a

f (z)dz

b) Definition of Fz0 (u) =∫ u

z0f (z)dz , u ∈ C

Fz0 (u) is independent of the path from z0 to u included in DFz0 (u) is a primitive of f (z) such that F ′z0

(u) = f (u).


Residue theorem

Outline

Some Generalities

Usual functions



Residue theoremTheorem for a bounded domain DApplication to integral calculusApplication to the sum of a series

Laplace transform

Z transform


Residue theorem

Theorem for a bounded domain D

Outline

Some Generalities

Usual functions




Laplace transform

Z transform


Residue theorem


Residue theorem

Assumptions

I f holomorphic on Ω\ ∪jzj , Ω non-empty open set of C

I zj isolated singularities of f

I D ⊂ Ω 1-connected domain of contour ∂D inside Ω

Conclusion ∫∂D+ f (z)dz = 2iπ

∑zj∈D

resf (zj)

with (definition of resf (zj)) :

resf (zj) = limr→0

12iπ

∫C+(zj ,r)

f (z)dz


Residue theorem


Remarks and definition

I Isolated singularities (IS, or isolated singular point)zj is an IS of f (z) if and only if ∃r > 0 such that f is holomorphic ind(zj , r)\ zj, where d(zj , r) stands for the disc of center zi andradius ri .

I Computing the residue thanks to the Laurent seriesIf zj is an IS, one admits that f has a Laurent series in d(zj , r)\zj :

f (z) =∞∑n=1

bn(z − zj)n

+∞∑n=0

an(z − zj)n

Thus, it comes:∫C+(zj ,r)

f (z)dz =∞∑n=1

∫C+

bn(z − zj)n

dz +∞∑n=0

∫C+

an(z − zj)ndz


Residue theorem



We set z − zj = re iθ and it yields

∞∑n=1

∫ 2π

0

bnidθ

rn−1e i(n−1)θ+ i

∞∑n=0

∫ 2π

0

anrn+1e i(n+1)θdθ

All the integrals are null (straightforward...) except:∫ 2π

0

bnidθ

rn−1e i(n−1)θwith n = 1

Thus : ∫C+(zj ,r)

f (z)dz =

∫ 2π

0

b1idθ = 2iπb1

Conclusion : resf (zj) is the coefficient of the term 1z−zj of the main

part of the Laurent series of f .


Residue theorem


Remarks and definitionI Computing the residue in case of a pole of order p

One computes the Taylor series of ϕ(z) = (z − zj)pf (z) which is

holomorphic in V (zj)

ϕ(z) = ϕ(zj) + ...+(z − zj)

p−1

(p − 1)!ϕ

(p−1)(zj )

+ ...

As a consequence, the Laurent series of f is:

f (z) =ϕ(zj)

(z − zj)p+ ...+

ϕ(p−1)(zj )

(p − 1)!(z − zj)+ ...

thus

resf (zj) = 1(p−1)!ϕ

(p−1)(zj )

= 1(p−1)!

dp−1

dzp−1 [(z − zj)pf (z)]

∣∣∣z=zj

In practice:I for p > 2, one compute the Laurent series,I for p = 2, one can use resf (zj) = d

dz (z − zj)2f (z)

∣∣z=zj

,

I for p = 1, one has resf (zj) = limz→zj

(z − zj)f (z)


Residue theorem



Interesting particular case : zj pole of order 1, f (z) = P(z)Q(z) , P(zj) 6= 0

One expands Q(z) :

Q(z) = 0 + (z − zj)Q′(zj) +

(z − zj)2

2!Q ′′(zj) + ...

thus

limz→zj

(z − zj)f (z) =P(zj)

Q ′(zj)

This formula is interesting for some residue calculus, such as f (z) = 1sin z

en z = 0. Indeed:

resf (0) =P(0)

Q ′(0)=

1

cos 0= 1


Residue theorem

Application to integral calculus

Outline

Some Generalities

Usual functions




Laplace transform

Z transform


Residue theorem


Integrals of the form: I =∫∞−∞ f (x)dx

Very often, one defines f (z) and the contour which consists of a straightline associated with I and a circular parts which close path. Example:

Computing

I =

∫ +∞

−∞

x2 + 1

x4 + 1dx


Residue theorem


Integrals defined by a multifunction

Example: show that, for a ∈ ]0, 1[

J =

∫ ∞0

xa−1

1 + xdx =

π

sin (πa)


Residue theorem


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 θand sin θ as functions of z .It consists of computing an integral on the unit circle.Example : show that

J =

∫ 2π

0

dθ

5 + 3 sin θ=π

2


Residue theorem

Application to the sum of a series

Outline

Some Generalities

Usual functions




Laplace transform

Z transform


Residue theorem



See exercise session.


Laplace transform

Outline

Some Generalities

Usual functions



Residue theorem

Laplace transformDefinitionPropertiesInverse Laplace transformApplications

Z transform


Laplace transform

Definition

Outline

Some Generalities

Usual functions



Residue theorem


Z transform


Laplace transform

Definition

Definition

Set of the (Laplace) transformable functionsE is the set of the functions f defined on R+ such that• f is locally integrable, i.e.,

∫ A

0f (t)dt <∞,∀A

• It exists x0 such that∫∞

0e−x0t f (t)dt <∞

Laplace transformFor f ∈ E , one defines its Laplace transform as

F (p) ,∫∞

0e−pt f (t)dt p ∈ C

Notation: F (p) = TL(f (t))


Laplace transform

Definition

DefinitionConvergences

(simple) Convergence

Thereom 1If F (p) exists for p = p0 = x0 + iy0 thenF (p) exists ∀p such that Rep > Rep0 = x0

Consequence : x ∈ R,F (p) <∞ admits a lower bound denoted xc andcalled abscissa of (simple) convergence of F .

Absolute convergence

Theorem 2If∫∞

0

∣∣e−pt f (t)∣∣ dt exists for p = p0 = x0 + iy0 then∫∞

0

∣∣e−pt f (t)∣∣ dt exists ∀p such that Rep > Rep0 = x0

Consequence :x ∈ R,

∫∞0

∣∣e−pt f (t)∣∣ dt <∞ admits a lower bound denoted

xca and called abscissa of absolute convergence of F (obviously, xc ≤ xca)

Example: f (t) = ekt sin[ekt], k > 0, xc = 0 and xca = k.

Remark : one often has xc = xca.


Laplace transform

Definition

Definition

Fundamental theoremIf f (t) is piecewise continuous on R+,then F (p) =

∫∞0

e−pt f (t)dt is holomorphic on ]xc ,+∞[ andthen it is infinitely differentiable on ]xc ,+∞[ with

dnF (p)dpn =

∫∞0

dn

dpn [e−pt f (t)] dt

Consequence: deriving xc from F (p)

If F (p) a function of the complex variable p is the Laplace transformof a function f (t) which admits isolated singularities skand branching points rj in C, then xc = supRe(sk , rj)

Examples: F (p) = 1p(p−2) xc = 2

F (p) = 1p+1 xc = 0


Laplace transform

Properties

Outline

Some Generalities

Usual functions



Residue theorem


Z transform


Laplace transform

Properties

Usual properties

a) LinearityTL (λf + µg) =λF (p) + µG (p)

Generally, abscissa of convergence xc = sup(xcf , xcg ).b) Derivation* with respect to p

TL (−1)ntnf (t) =dn

dpnF (p)

* with respect to t (f continuous on [0,+∞[)

TL [f ′(t)] = pF (p)− f (0+)

Generalization:

TL[f (n)(t)

]= pnF (p)− pn−1f (0+)− ...− f (n−1)(0+)

Application: resolution of linear differential equations


Laplace transform

Properties

Usual properties

c) Integration* LT of a primitive

TL

[∫ t

0

f (u)du

]=

F (p)

p

Abscissa of convergence: sup(xc , 0)* Primitive of a LT

TL

[f (t)

t

]=

∫ ∞p

F (u)du


Laplace transform

Properties

Usual properties

d) Translation* with respect to p

TL[eat f (t)

]= F (p − a)

Abscissa of convergence: xc + Re(a)* with respect to t

TL [f (t − a)U(t − a)] = e−apF (p)

Abscissa of convergence: xcRemark: Application to periodic functionse) Scaling

TL[f( tk

)]= kF (kp) k > 0

Abscissa of convergence: xck


Laplace transform

Properties

Usual properties

f) Convolution

TL

[∫ t

0

f (u)g(t − u)du

]= F (p)G (p)

g) Theorems of the initial and final values

limt→0+

f (t) = limp→∞

pF (p)

limt→∞

f (t) = limp→0

pF (p)

h) Transform of seriesSeries of general term an

tn

n!with abscissa of convergence Rc =∞

TL[∑∞

n=1 antn

n!

]=∑∞

n=1an

pn+1

Example: show that TL[

sinωtt

]= Arctg ωp

Use two methods: series expansion and TL[x(t)t

]Prof. Nicolas Dobigeon Complex variables - LT & ZT 79 / 105

Laplace transform

Properties

Some Laplace transforms

Function TL ConvergenceU(t) 1

p xc = 0

eαt 1p−α xc = Reα

e iωt 1p−iω xc = 0

ch (αt) pp2−α2 xc = supRe(α,−α)

sh (αt) αp2−α2 xc = supRe(α,−α)

cosωt pp2+ω2 xc = 0

sinωt ωp2+ω2 xc = 0

t 1p2 xc = 0

tn, n ∈ N n!pn+1 xc = 0

tα, α ∈ R Γ(α+1)pα+1

with Γ(x) =∫∞

0e−ttx−1dt et Γ(n + 1) = nΓ(n) = n!


Laplace transform

Inverse Laplace transform

Outline

Some Generalities

Usual functions



Residue theorem


Z transform


Laplace transform

Inverse Laplace transform

Inversion formula

X (p) =

∫ ∞0

x(t)e−ptdt =

∫ ∞0

x(t)e−ate−j2πftdt

with p = a + j2πfAnalogy with the Fourier transform

X (f ) = TF (x(t)) =∫R x(t)e−j2πft

x(t) = TF−1(X (f )) =∫R X (f )e+j2πftdf

Hence :

X (p) = TF[x(t)e−atU(t)

]and thus the inversion formula:

x(t)U(t) = 12iπ

∫D↑ X (p)eptdp

One applies the residue theorem with X (p)ept .Example: X (p) = 1√

p


Laplace transform

Applications

Outline

Some Generalities

Usual functions



Residue theorem


Z transform


Laplace transform

Applications

Differential equations with constant coefficients

y (n) + a1y(n−1) + ...+ any(t) = f (t)

Initial conditions

y(0) = b0, y′(0) = b1, ..., y

(n−1)(0) = bn−1

Laplace transform

TL [any(t)] = anY (p)TL[y (n)(t)

]= pnY (p)− pn−1y(0+)− ...− y (n−1)(0+)

TL [Ωn(y)] = Ωn(p)Y (p)− Qn−1(p)TL [f (t)] = F (p)

Algebraic problem

Ωn(p)Y (p) = Qn−1(p) + F (p)

Y (p) = Qn−1(p)Ωn(p) + F (p)

Ωn(p) = Y1(p) + Y2(p)


Laplace transform

Applications

Differential equations with constant coefficients

a) Y1(p) Algebraic fraction

Y1(p) =Qn−1(p)∏r

i=1(p − pi )ki

where pi is a ki -order root with∑r

i=1 ki = nPartial fraction decomposition :

Y1(p) =r∑

i=1

Ai1

p − pi+

Ai2

(p − pi )2 + ...+

Aiki

(p − pi )ki

where

y1(t) =∑r

i=1 epi t[Ai1 + Ai2t + ...+ Aiki t

ki−1]

b) Y2(p) = F (p)Ωn(p) = F (p)× 1

Ωn(p) thus:

y2(t) =

∫ t

0

f (u)Rn(t − u)du

Hence, the solution of the problem is y(t) = y1(t) + y2(t)Prof. Nicolas Dobigeon Complex variables - LT & ZT 85 / 105

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)

∂2f∂x2 − 1

c2∂2f∂t2 = 0

Initial conditions

f (x , 0) = ϕ(x)∂f

∂t(x , 0) = ψ(x)

Conditions at limits

f (∞, t) = 0f (0, t) = g(t)


Laplace transform

Applications


Solution thanks to the LT (p is a considered as a parameter)

F (x , p) =

∫ ∞0

e−pt f (x , t)dt

TL

[∂f

∂t

]= pF (x , p)− f (x , 0)

= pF (x , p)− ϕ(x)

TL

[∂2f

∂t2(x , t)

]= p2F (x , p)− pf (x , 0)− ∂f

∂t(x , 0)

= p2F (x , p)− pϕ(x)− ψ(x)

TL

[∂2f

∂x2(x , t)

]=

∫ ∞0

e−pt∂2f (x , t)

∂x2dt

=∂2

∂x2

∫ ∞0

e−pt f (x , t)dt =d2F (x , p)

dx2


Laplace transform

Applications


One obtains

d2F (x,p)dx2 − p2F (x , p) = pϕ(x) + ψ(x)

with

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).


Z transform

Outline

Some Generalities

Usual functions



Residue theorem

Laplace transform

Z transformDefinitionPropertiesInverse Z transformApplicationsLaplace and Z transforms


Z transform

Definition

Outline

Some Generalities

Usual functions



Residue theorem

Laplace transform



Z transform

Definition

Definition

DefinitionOne defines the Z transform of a series x(n), n ∈ Z as:

X (z) =+∞∑

n=−∞

x(n)z−n z ∈ C

Notation:X (z) = TZ(x(n))

Remark: bilateral and unilateral TZ.


Z transform

Definition

Definition

Domain of convergenceThe domain of convergence is the set of complex numbers z such that theseries X (z) converges.

Reminder: Cauchy criterion

limn→+∞

n√|un| < 1 =⇒

+∞∑n=0

un converge

One has a sufficient condition of convergence. Thanks to this criterion, oneshows that the series X (z) converges once:

0 ≤ R−x < |z | < R+x ≤ +∞

Example: X (z) =∑+∞

n=0 z−n converges for |z | > 1


Z transform

Properties

Outline

Some Generalities

Usual functions



Residue theorem

Laplace transform



Z transform

Properties

Usual properties

LinearityTZ (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 − n0)) = z−n0X (z)

Same domain of convergence as X (z).Scaling

TZ (anx(n)) = X(za

)Domain of convergence: |a|R−x < |z | < |a|R+

x


Z transform

Properties

Usual properties

DifferentiabilityThe Z transform defines a Laurent series which is infinitely differentiableterm-by-term in its domain of convergence. Thus

TZ (nx(n)) = −z dX (z)

dz

Same domain of convergence as X (z).Convolution productThe convolution between the series x(n) and y(n) is defined as:

u(n) = x(n) ∗ y(n) =+∞∑

k=−∞

x(k)y(n − k)

ThusTZ (x(n) ∗ y(n)) = X (z)Y (z)

The domain of convergence of U(z) can be larger than the intersectionof domains of convergence of X (z) and Y (z), respectively.


Z transform

Inverse Z transform

Outline

Some Generalities

Usual functions



Residue theorem

Laplace transform



Z transform

Inverse Z transform

Inverse Z transform

The inverse Z transform is given by:

x(n) =1

j2π

∫C+

X (z)zn−1dz

where C is a closed path included into the domain of convergence


Z transform

Inverse Z transform

TZ inverse

ProofOne has to compute the integrals

J(n, k) =

∫C+

zn−k−1dz

Thanks to the residue theorem, one shows that:

J(n, k) =

0 si n 6= kj2π si n = k

Hence:

1

j2π

∫C+

X (z)zn−1dz =1

j2π

∫C+

( ∞∑k=−∞

x(k)z−k

)zn−1dz

=1

j2π

∞∑k=−∞

x(k)J(n, k)

= x(n)

Remark : tablesProf. Nicolas Dobigeon Complex variables - LT & ZT 98 / 105

Z transform

Applications

Outline

Some Generalities

Usual functions



Residue theorem

Laplace transform



Z transform

Applications

Discrete signal filtering

See exercise session and/or later.


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) = bnU(n) with |b| < 1

where U(n) is the Heaviside step function.

I Compute y(n) for n ≥ 0 given that y(n) = 0 for n < 0.

I Determine the impulse response of the system h(n) such thaty(n) = x(n) ∗ h(n).


Z transform

Laplace and Z transforms

Outline

Some Generalities

Usual functions



Residue theorem

Laplace transform



Z transform



Let x(t) define a causal signal whose Laplace transform is:

X (p) =

∫ ∞0

x(t)e−ptdt

One samples this signal with period T and one denotes X (z) its Ztransform:

X (z) =∞∑n=0

x(nT )z−n

ThenX (z) =

∑res X (p)

1−epT z−1


Z transform



The formula of inverse Laplace transform provides

x(t)U(t) =1

2iπ

∫D↑

X (p)eptdp

hence

X (z) =∞∑n=0

x(nT )z−n =∞∑n=0

[1

2iπ

∫D↑

X (p)epnTdp

]z−n

=1

2iπ

∫D↑

X (p)∞∑n=0

(z−1epT

)ndp

Once∣∣z−1epT

∣∣ < 1, on a

X (z) =1

2iπ

∫D↑

X (p)1

1− z−1epTdp =

∑res

X (p)

1− epT z−1


Z transform


Complex VariablesLaplace Transform – Z Transform

Prof. Nicolas Dobigeon

University of ToulouseIRIT/INP-ENSEEIHT

http://www.enseeiht.fr/[email protected]


http://www.enseeiht.fr/~dobigeon

complex variables - laplace transform z transform

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