Complex VariablesComplex Variables

ForFor

ECON 397 ECON 397 MacroeconometricsMacroeconometrics

Steve CunninghamSteve Cunningham

Open Disks or NeighborhoodsOpen Disks or Neighborhoods

Definition. The set of all points z which Definition. The set of all points z which satisfy the inequality |satisfy the inequality |z z –– zz00|<|<ρρ, where , where ρρ is is a positive real number is called an a positive real number is called an open diskopen diskor or neighborhood of zneighborhood of z0 0 . .

Remark. The Remark. The unit diskunit disk, i.e., the neighborhood , i.e., the neighborhood ||zz|< 1, is of particular significance. |< 1, is of particular significance.

1

Interior PointInterior Point

Definition. A point is called an Definition. A point is called an interior interior point of S point of S if and only if there exists at least one if and only if there exists at least one neighborhood of neighborhood of zz00 which is completely which is completely contained in contained in SS. .

Sz ∈0

z0

S

Open Set. Closed Set.Open Set. Closed Set.

Definition. If every point of a set Definition. If every point of a set SS is an interior is an interior point of point of SS, we say that , we say that SS is an is an open setopen set. .

Definition. If Definition. If B(S) B(S) ⊂⊂ S, S, i.e., if i.e., if SS contains all of its contains all of its boundary points, then it is called a boundary points, then it is called a closed setclosed set. .

Sets may be neither open nor closed.Sets may be neither open nor closed.

Open ClosedNeither

ConnectedConnected

An open set An open set SS is said to be is said to be connected connected if every pair if every pair of points of points zz11 andand zz22 in in S S can be joined by a can be joined by a polygonal line that lies entirely in polygonal line that lies entirely in SS. Roughly . Roughly speaking, this means that speaking, this means that SS consists of a “single consists of a “single piece”, although it may contain holes. piece”, although it may contain holes.

SS

z1 z2

Domain, Region, Closure, Domain, Region, Closure, Bounded, CompactBounded, Compact

An open, connected set is called a An open, connected set is called a domain. domain. A A regionregionis a domain together with some, none, or all of its is a domain together with some, none, or all of its boundary points. The boundary points. The closure closure of a set of a set SS denoted , is denoted , is the set of the set of SS together with all of its boundary. Thustogether with all of its boundary. Thus

..

A set of points A set of points SS is is bounded bounded if there exists a positive if there exists a positive real number real number RR such that |such that |zz|<|<RR for every for every z z ∈∈ S.S.

A region which is both closed and bounded is said A region which is both closed and bounded is said to be to be compact.compact.

S

)(SBSS ∪=

Review: Real Functions of Real Review: Real Functions of Real VariablesVariables

Definition. Let Definition. Let ΓΓ ⊂⊂ ℜℜ. A . A function ffunction f is a rule which is a rule which assigns to each element assigns to each element aa ∈∈ ΓΓ one and only one one and only one element element b b ∈∈ ΩΩ, , ΩΩ ⊂⊂ ℜℜ. We write . We write f: f: Γ→Γ→ ΩΩ, or in , or in the specific case the specific case b = f(a)b = f(a), and call , and call b b ““the image of the image of aaunder under ff..””We call We call ΓΓ ““the domain of definition of the domain of definition of f f ”” or or simply simply ““the domain of the domain of f f ””. . We call We call ΩΩ ““the range of the range of f.f.””We call the set of all the We call the set of all the imagesimages of of ΓΓ, denoted , denoted f f ((ΓΓ), ), the image of the function the image of the function ff . We alternately call . We alternately call f f a a mapping mapping from from ΓΓ to to ΩΩ..

Real FunctionReal Function

In effect, a function of a real variable maps In effect, a function of a real variable maps from one real line to another.from one real line to another.

Γ Ω

f

Complex FunctionComplex Function

Definition. Complex function of a complex Definition. Complex function of a complex variable. Let variable. Let ΦΦ ⊂⊂ C. A C. A function f function f defined on defined on ΦΦ is a rule which assigns to each is a rule which assigns to each zz ∈∈ ΦΦ a a complex number complex number w. w. The number The number ww is called is called a a value of f at z value of f at z and is denoted by and is denoted by f(z)f(z), i.e., , i.e., w = f(z). w = f(z). The set The set ΦΦ is called the is called the domain of definition of f. domain of definition of f. Although the domain of definition is often Although the domain of definition is often a domain, it need not be.a domain, it need not be.

RemarkRemark

Properties of a realProperties of a real--valued function of a real valued function of a real variable are often exhibited by the graph of the variable are often exhibited by the graph of the function. But when function. But when w = f(z)w = f(z), where , where zz and and ww are are complex, no such convenient graphical complex, no such convenient graphical representation is available because each of the representation is available because each of the numbers numbers zz and and ww is located in a plane rather than is located in a plane rather than a line. a line. We can display some information about the We can display some information about the function by indicating pairs of corresponding function by indicating pairs of corresponding points points z = z = ((x,yx,y) and ) and w = w = ((u,vu,v). To do this, it is ). To do this, it is usually easiest to draw the usually easiest to draw the z z and and w w planes planes separately.separately.

Graph of Complex FunctionGraph of Complex Function

x u

y v

z-plane w-plane

domain ofdefinition

range

w = f(z)

Example 1Example 1

Describe the range of the function f(z) = x2 + 2i, defined on (the domain is) the unit disk |z|≤ 1.

Solution: We have u(x,y) = x2 and v(x,y) = 2. Thus as z varies over the closed unit disk, u varies between 0 and 1, and v is constant (=2).

Therefore w = f(z) = u(x,y) + iv(x,y) = x2 +2i is a line segment from w = 2i to w = 1 + 2i.

x

y

u

vf(z)

domain

range

Example 2Example 2

Describe the function f(z) = z3 for z in the semidisk given by |z|≤ 2, Im z ≥ 0.

Solution: We know that the points in the sector of the semidisk from Arg z = 0 to Arg z = 2π/3, when cubed cover the entire disk |w|≤ 8 because

The cubes of the remaining points of z also fall into this disk, overlapping it in the upper half-plane as depicted on the next screen.

( ) iiee ππ 2

33

282 =⎥⎦

⎤⎢⎣⎡

2-2 x

y

u

v

8-8

-8

8

2

w = z3

SequenceSequence

Definition. A Definition. A sequence of complex numberssequence of complex numbers, denoted , denoted , is a function , is a function ff, such that , such that f: f: N N →→ C, i.e, it is a C, i.e, it is a

function whose domain is the set of natural function whose domain is the set of natural numbers between 1 and numbers between 1 and kk, and whose range is a , and whose range is a subset of the complex numbers. If subset of the complex numbers. If k = k = ∞∞, then the , then the sequence is called sequence is called infinite infinite and is denoted by , and is denoted by , or more often, or more often, zznn .. (The notation (The notation f(n) f(n) is equivalent.)is equivalent.)

Having defined sequences and a means for Having defined sequences and a means for measuring the distance between points, we measuring the distance between points, we proceed to define the limit of a sequence.proceed to define the limit of a sequence.

knz 1

∞1nz

Limit of a SequenceLimit of a Sequence

Definition. A sequence of complex numbers Definition. A sequence of complex numbers is said to have the limit is said to have the limit zz0 0 , , or to converge to or to converge to zz00 , if , if for any for any εε > 0, there exists an integer > 0, there exists an integer NN such that such that ||zznn –– zz00| < | < εε for all for all nn > > NN. We denote this by . We denote this by

Geometrically, this amounts to the fact that Geometrically, this amounts to the fact that zz00 is is the only point of the only point of zznn such that any neighborhood such that any neighborhood about it, no matter how small, contains an infinite about it, no matter how small, contains an infinite number of points number of points zznn ..

∞1nz

.

lim

0

0

∞→→

=∞→

naszz

orzz

n

nn

Limit of a FunctionLimit of a Function

We say that the complex number We say that the complex number ww00 is the limit of is the limit of the function the function f(z) f(z) as as z z approaches approaches zz00 if if f(z) f(z) stays close stays close to to ww00 whenever whenever zz is sufficiently near is sufficiently near zz00 . Formally, . Formally, we state:we state:

Definition. Limit of a Complex Sequence. Let Definition. Limit of a Complex Sequence. Let f(z) f(z) be a function defined in some neighborhood of be a function defined in some neighborhood of zz00

except with the possible exception of the point except with the possible exception of the point zz00

is the number is the number ww00 if for any real number if for any real number εε > 0 > 0 there exists a positive real number there exists a positive real number δδ > 0 such that > 0 such that ||f(z) f(z) –– ww00|< |< εε whenever 0<|whenever 0<|zz -- zz00|< |< δδ..

Limits: InterpretationLimits: Interpretation

We can interpret this to mean that if we observe points zwithin a radius δδ of of zz00, we can find a corresponding disk , we can find a corresponding disk about about ww00 such that all the points in the disk about such that all the points in the disk about zz00 are are mapped into it. That is, mapped into it. That is, any any neighborhood of neighborhood of ww00 contains contains all the values assumed by all the values assumed by ff in some full neighborhood of in some full neighborhood of zz00, except possibly , except possibly f(zf(z00)). .

zz00δδ w0

εw = f(z)

z-plane w-planeu

v

x

y

Properties of LimitsProperties of Limits

If as If as zz →→ zz00, , lim lim f(z)f(z) →→ A and A and lim lim g(z)g(z) →→ B, B, thenthen

limlim [ [ f(z) f(z) ±± g(z) g(z) ] = A ] = A ±± BB

limlim f(z)g(z)f(z)g(z) = AB, and= AB, and

limlim f(z)/g(z) f(z)/g(z) = A/B. if B = A/B. if B ≠≠ 0.0.

ContinuityContinuity

Definition. Let Definition. Let f(z) f(z) be a function such that be a function such that f: f: C C. C C. We call We call f(z) continuous at zf(z) continuous at z00 iffiff::

FF is defined in a neighborhood of zis defined in a neighborhood of z00,,The limit exists, andThe limit exists, and

A function A function f f is said to be is said to be continuous on a set Scontinuous on a set S if if it is continuous at each point of S. If a it is continuous at each point of S. If a function is not continuous at a point, then it function is not continuous at a point, then it is said to be singular at the point.is said to be singular at the point.

)()(lim 00

zfzfzz

=→

Note on ContinuityNote on Continuity

One can show that One can show that f(z)f(z) approaches a limit approaches a limit precisely when its real and imaginary parts precisely when its real and imaginary parts approach limits, and the continuity of approach limits, and the continuity of f(z) f(z) is is equivalent to the continuity of its real and equivalent to the continuity of its real and imaginary parts.imaginary parts.

Properties of Continuous FunctionsProperties of Continuous Functions

If If f(z) f(z) and and g(z) g(z) are continuous at are continuous at zz00, then so , then so are are f(z) f(z) ±± g(z) g(z) and and f(z)g(z). f(z)g(z). The quotient The quotient f(z)/g(z)f(z)/g(z)is also continuous at is also continuous at zz00 provided that provided that g(zg(z00) ) ≠≠ 0.0.

Also, continuous functions map compact Also, continuous functions map compact sets into compact sets. sets into compact sets.

DerivativesDerivatives

Differentiation of complexDifferentiation of complex--valued functions is valued functions is completely analogous to the real case:completely analogous to the real case:

Definition. Derivative. Let Definition. Derivative. Let f(z) f(z) be a complexbe a complex--valued function defined in a neighborhood of valued function defined in a neighborhood of zz00. . Then the Then the derivative derivative of of f(z) f(z) at at zz00 is given byis given by

Provided this limit exists. Provided this limit exists. F(z) F(z) is said to be is said to be differentiable at zdifferentiable at z00..

zzfzzf

zfz ∆

−∆+=′

→∆

)()(lim)( 00

00

Properties of DerivativesProperties of Derivatives

( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( )[ ]

( )

( )[ ] ( )[ ] ( ) Rule. Chain''

.0if,''

'

'''

. constant anyfor ''

'''

000

020

00000

00000

00

000

zgzgfzgfdzd

zgzg

zgzfzfzgz

gf

zgzfzgzfzfg

czcfzcf

zgzfzgf

=

≠−

=⎟⎠

⎞⎜⎝

⎛

+==

±=±

Analytic. Analytic. HolomorphicHolomorphic..

Definition. A complexDefinition. A complex--valued function valued function ff ((zz) is said ) is said to be to be analyticanalytic, or equivalently, , or equivalently, holomorphicholomorphic, on an , on an open set open set ΩΩ if it has a derivative at every point of if it has a derivative at every point of ΩΩ. (The term . (The term ““regularregular”” is also used.)is also used.)It is important that a function may be It is important that a function may be differentiable at a single point only. Analyticity differentiable at a single point only. Analyticity implies differentiability within a neighborhood of implies differentiability within a neighborhood of the point. This permits expansion of the function the point. This permits expansion of the function by a Taylor series about the point.by a Taylor series about the point.If If ff ((zz) is analytic on the whole complex plane, ) is analytic on the whole complex plane, then it is said to be an then it is said to be an entire functionentire function. .

Rational Function.Rational Function.

Definition. If Definition. If ff and and gg are polynomials in are polynomials in zz, , then then h h ((zz) = ) = ff ((zz)/)/gg((zz), ), gg((zz) ) ≠≠ 0 is called a 0 is called a rational function. rational function. Remarks.Remarks.

All polynomial functions of All polynomial functions of zz are entire.are entire.A rational function of A rational function of zz is analytic at every is analytic at every point for which its denominator is nonzero.point for which its denominator is nonzero.If a function can be reduced to a polynomial If a function can be reduced to a polynomial function which does not involve , then it is function which does not involve , then it is analytic. analytic.

z

Example 1Example 1

11

11

)(

21

2

21

2)(

2,

2

)1(

1)(

1

221

221

−=

+−−−

=

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+

−−−+

=

−=

+=

+−−−

=

zzzzzz

zf

izzzz

izz

izz

zf

izz

yzz

xlet

yxiyx

zf

Thus f1(z) is analytic at all points except z=1.

Example 2Example 2

13)(2

312

322

)(

2,

2

313)(

2

22

2

222

++=

⎟⎠⎞

⎜⎝⎛ −++⎟

⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ +=

−=

+=

++++=

zzzzfizz

izz

izzzz

zf

izz

yzz

xlet

yixyxzf

Thus f2(z) is nowhere analytic.

Testing for AnalyticityTesting for Analyticity

Determining the analyticity of a function by searching for in its expression that cannot be removed is at best awkward. Observe:

z

( ) ( )5523

5345 1)(

zzzz

zzzzzzzf

+−

−+−+=

It would be difficult and time consuming to try to reduce this expression to a form in which you could be sure that the could not be removed. The method cannot be used when anything but algebraic functions are used.

z

CauchyCauchy--Riemann Equations (1)Riemann Equations (1)

If the function f (z) = u(x,y) + iv(x,y) is differentiable at z0 = x0 + iy0, then the limit

zzfzzf

zfz ∆

−∆+=′

→∆

)()(lim)( 00

00

can be evaluated by allowing ∆z to approach zero from any direction in the complex plane.

CauchyCauchy--Riemann Equations (2)Riemann Equations (2)If it approaches along the x-axis, then ∆z = ∆x, and we obtain

[ ] [ ]x

yxivyxuyxxivyxxuzf

x ∆+−∆++∆+=

→∆

),(),(),(),(lim)(' 00000000

00

⎥⎦⎤

⎢⎣⎡

∆−∆+

+⎥⎦⎤

⎢⎣⎡

∆−∆+

=→∆→∆ x

yxvyxxvi

xyxuyxxu

zfxx

),(),(lim

),(),(lim)(' 0000

0

0000

00

But the limits of the bracketed expression are just the first partial derivatives of u and v with respect to x, so that:

).,(),()(' 00000 yxxv

iyxxu

zf∂∂

+∂∂

=

CauchyCauchy--Riemann Equations (3)Riemann Equations (3)If it approaches along the y-axis, then ∆z = ∆y, and we obtain

⎥⎦

⎤⎢⎣

⎡∆

−∆+=→∆ y

yxuyyxuzf

y

),(),(lim)(' 0000

00

And, therefore

).,(),()(' 00000 yxyv

yxyu

izf∂∂

+∂∂

−=

⎥⎦

⎤⎢⎣

⎡∆

−∆++

→∆ yyxvyyxv

iy

),(),(lim 0000

0

CauchyCauchy--Riemann Equations (4)Riemann Equations (4)

By definition, a limit exists only if it is unique. Therefore, these two expressions must be equivalent. Equating real and imaginary parts, we have that

xv

yu

yv

xu

∂∂

−=∂∂

∂∂

=∂∂

and

must hold at z0 = x0 + iy0 . These equations are called the Cauchy-Riemann Equations. Their importance is made clear in the following theorem.

CauchyCauchy--Riemann Equations (5)Riemann Equations (5)

Theorem. Let Theorem. Let ff ((zz) = ) = uu((x,yx,y) + ) + iviv((x,yx,y) be ) be defined in some open set defined in some open set ΓΓ containing the containing the point point zz00. If the first partial derivatives of . If the first partial derivatives of u u and and vv exist in exist in ΓΓ, and are continuous at , and are continuous at zz0 0 , , and satisfy the Cauchyand satisfy the Cauchy--Riemann equations Riemann equations at at zz00, then , then ff ((zz) is differentiable at ) is differentiable at zz00. . Consequently, if the first partial Consequently, if the first partial derivatives are continuous and satisfy the derivatives are continuous and satisfy the CauchyCauchy--Riemann equations at Riemann equations at all all points of points of ΓΓ, then , then ff ((zz) is ) is analytic in analytic in ΓΓ..

Example 1Example 1

1,1,2,2

)()()( 22

−=∂∂

=∂∂

=∂∂

=∂∂

−++=

xv

yu

yyv

xxu

xyiyxzf

Hence, the Cauchy-Riemann equations are satisfied only on the line x = y, and therefore in no open disk. Thus, by the theorem, f (z) is nowhere analytic.

Example 2Example 2Prove that f (z) is entire and find its derivative.

yexv

yeyu

yeyv

yexu

yieyezf

xxxx

xx

sin,sin,cos,cos

:Solution

sincos)(

=∂∂

−=∂∂

=∂∂

=∂∂

+=

The first partials are continuous and satisfy the Cauchy-Riemann equations at every point.

.sincos)(' yieyexv

ixu

zf xx +=∂∂

+∂∂

=

Harmonic FunctionsHarmonic Functions

Definition. Harmonic. A realDefinition. Harmonic. A real--valued function valued function ϕϕ((xx,,yy) is said to be ) is said to be harmonic harmonic in a domain in a domain DD if all of if all of its secondits second--order partial derivatives are continuous order partial derivatives are continuous in in DD and if each point of and if each point of DD satisfiessatisfies

.02

2

2

2

=∂∂+

∂∂

yxϕϕ

Theorem. If f (z) = u(x,y) + iv(x,y) is analytic in a domain D, then each of the functions u(x,y) and v(x,y) is harmonic in D.

Harmonic ConjugateHarmonic Conjugate

Given a function Given a function uu((xx,,yy) harmonic in, say, an ) harmonic in, say, an open disk, then we can find another open disk, then we can find another harmonic function harmonic function vv((xx,,yy) so that ) so that u + ivu + iv is an is an analytic function of analytic function of zz in the disk. Such a in the disk. Such a function function vv is called a is called a harmonic conjugateharmonic conjugate of of u.u.

ExampleExampleConstruct an analytic function whose real part is:

.3),( 23 yxyxyxu +−=

Solution: First verify that this function is harmonic.

.066

6 and16

6and33

2

2

2

2

2

2

2

222

=−=∂∂

+∂∂

−=∂∂

+−=∂∂

=∂∂

−=∂∂

xxyu

xu

and

xyu

xyyu

xx

uyx

xu

Example, Example, ContinuedContinued

16)2(

33)1( 22

−=∂∂−

=∂∂

−=∂∂

=∂∂

xyyu

xv

andyxxu

yv

Integrate (1) with respect to y:

( )( )

)(3),()3(

33

33

32

22

22

xyyxyxv

yyxv

yyxv

ξ+−=

∂−=∂

∂−=∂

∫∫

Example, Example, ContinuedContinued

Now take the derivative of v(x,y) with respect to x:

).('6 xxyxv ξ+=

∂∂

According to equation (2), this equals 6xy – 1. Thus,

.3),(

.)(and,)(So

.1ly,Equivalent.1)('and

16)('6

32 CyyxyxvAnd

Cxxxx

xx

xyxxy

+−=

+−=∂−=∂

−=∂∂

−=

−=+

∫ ∫ ξξ

ξξ

ξ

Example, Example, ContinuedContinued

The desired analytic function f (z) = u + iv is:

( ) ( )Cxyyxiyxyxzf +−−++−= 3223 33)(

Complex ExponentialComplex Exponential

We would like the complex exponential to We would like the complex exponential to be a natural extension of the real case, with be a natural extension of the real case, with ff ((zz) = ) = eezz entire. We begin by examining entire. We begin by examining eezz = = eex+x+iyiy = = eexxeeiyiy..eeiyiy = = cos cos yy + + ii sin sin y y by Euler’s and by Euler’s and DeMoivre’s DeMoivre’s relations.relations.Definition. Complex Exponential Function. Definition. Complex Exponential Function. If If zz = = xx + + iyiy, then , then eezz = e= exx((cos cos yy + + i i sin sin yy).).That is, |That is, |eezz|= e|= exx and and arg earg ezz = = yy..

More on ExponentialsMore on Exponentials

Recall that a function Recall that a function ff is is oneone--toto--oneone on a set on a set SS if the if the equation equation ff ((zz11) = ) = ff ((zz22), where ), where zz11,, zz2 2 ∈∈ SS, implies , implies that that zz1 1 = = zz22. The complex exponential function is . The complex exponential function is notnot oneone--toto--one on the whole plane.one on the whole plane.

Theorem. A necessary and sufficient condition Theorem. A necessary and sufficient condition that that eezz = 1 is that = 1 is that zz = 2= 2kkππii, where , where kk is an integer. is an integer.

Also, a necessary and sufficient condition that Also, a necessary and sufficient condition that is that is that zz1 1 = = zz2 2 ++ 22kkππii, where , where kk is an is an

integer. Thus integer. Thus eezz is a periodic function.is a periodic function.

21 ee zz =