Chapter 3 Analytic Functions of a Complex Variablefrancisjosephcampena.weebly.com/uploads/1/7/8/6/17869691/complexbeamer3.pdfModealg The Derivative Analytic Functions Example: Show

Modealg

TheDerivative

AnalyticFunctions

Chapter 3Analytic Functions of a Complex Variable

An Introduction to Complex Analysis

Leonor Aquino-RuivivarMathematics Department

De La Salle University-Manila

Modealg

TheDerivative

AnalyticFunctions

The Derivative of a Complex Valued Function

Let f be a function defined in a domain D, and let z0 ∈ D.The derivative of f with respect to z at the point z0, isdefined by

df (z0)

dz= f ′(z0) = lim

z→z0

f (z) − f (z0)

z − z0

if this limit exists. In this case, we say that f is differentiableat z0.

Modealg

TheDerivative

AnalyticFunctions

Example: Show that the function w = f (z) = z2 isdifferentiable in the whole complex plane, andf ′(z) = 2z for every z ∈ C.Remark: If f is differentiable at z0, then f is continuousat z0.The converse of the preceding statement is false.Consider the function w = |z|2.Show that the function f (z) = Re z does not have aderivative at any point z0 in C.

Modealg

TheDerivative

AnalyticFunctions

Example: Show that the function w = f (z) = z2 isdifferentiable in the whole complex plane, andf ′(z) = 2z for every z ∈ C.Remark: If f is differentiable at z0, then f is continuousat z0.The converse of the preceding statement is false.Consider the function w = |z|2.Show that the function f (z) = Re z does not have aderivative at any point z0 in C.

Modealg

TheDerivative

AnalyticFunctions

Example: Show that the function w = f (z) = z2 isdifferentiable in the whole complex plane, andf ′(z) = 2z for every z ∈ C.Remark: If f is differentiable at z0, then f is continuousat z0.The converse of the preceding statement is false.Consider the function w = |z|2.Show that the function f (z) = Re z does not have aderivative at any point z0 in C.

Modealg

TheDerivative

AnalyticFunctions

Example: Show that the function w = f (z) = z2 isdifferentiable in the whole complex plane, andf ′(z) = 2z for every z ∈ C.Remark: If f is differentiable at z0, then f is continuousat z0.The converse of the preceding statement is false.Consider the function w = |z|2.Show that the function f (z) = Re z does not have aderivative at any point z0 in C.

Modealg

TheDerivative

AnalyticFunctions

Example: Show that the function w = f (z) = z2 isdifferentiable in the whole complex plane, andf ′(z) = 2z for every z ∈ C.Remark: If f is differentiable at z0, then f is continuousat z0.The converse of the preceding statement is false.Consider the function w = |z|2.Show that the function f (z) = Re z does not have aderivative at any point z0 in C.

Modealg

TheDerivative

AnalyticFunctions

Theorems on Differentiability

Let D be a domain in C.(a) If f is a constant function in D, then f ′(z) = 0 ∀z ∈ D.(b) If f is the identity function in D, then f ′(z) = 1 ∀z ∈ D.(c) If n is a positive integer, then if f (z) = zn, then

f ′(z) = nzn−1 for every z ∈ D.

Modealg

TheDerivative

AnalyticFunctions

Theorems on Differentiability

Let D be a domain in C.(a) If f is a constant function in D, then f ′(z) = 0 ∀z ∈ D.(b) If f is the identity function in D, then f ′(z) = 1 ∀z ∈ D.(c) If n is a positive integer, then if f (z) = zn, then

f ′(z) = nzn−1 for every z ∈ D.

Modealg

TheDerivative

AnalyticFunctions

Theorems on Differentiability

Let D be a domain in C.(a) If f is a constant function in D, then f ′(z) = 0 ∀z ∈ D.(b) If f is the identity function in D, then f ′(z) = 1 ∀z ∈ D.(c) If n is a positive integer, then if f (z) = zn, then

f ′(z) = nzn−1 for every z ∈ D.

Modealg

TheDerivative

AnalyticFunctions

Theorems on Differentiability

Let D be a domain in C.(a) If f is a constant function in D, then f ′(z) = 0 ∀z ∈ D.(b) If f is the identity function in D, then f ′(z) = 1 ∀z ∈ D.(c) If n is a positive integer, then if f (z) = zn, then

f ′(z) = nzn−1 for every z ∈ D.

Modealg

TheDerivative

AnalyticFunctions

Differentiation Formulas

Let f , g be differentiable at a number z0. Then f + g, f − g,and fg are also differentiable at z0, and f/g is differentiableat z0 if g(z0) 6= 0. Moreover,

Dz(f ± g)(z0) = f ′(z0) ± g′(z0)

Dz(fg)(z0) = f (z0)g′(z0) + g(z0)f ′(z0)

Dz

(fg

)(z0) =

g(z0)f ′(z0) − g(z0)f ′(z0)

(g(z0))2

Modealg

TheDerivative

AnalyticFunctions

The Chain Rule

Suppose η = g(z) is differentiable in a domain D, and itsrange lies in a domain R, and a second function w = f (η) isdifferentiable in the domain R with range lying in a domainS, then the composite function w = f (g(z)) is differentiableat each z ∈ D, and

dwdz

=dwdη

· dη

dz

Modealg

TheDerivative

AnalyticFunctions

Analytic Functions

A function f is said to be analytic in a domain D if f hasa derivative at every point in D.A function f is analytic at a point z0 if there exists anopen disk B(z0, ε) such that f ′(z) exists for everyz ∈ B(z0, ε).A function f is analytic in a domain D if it is analytic atevery point in D.A function f is analytic in a set S if it is analytic a t everyelement of S.A function which is analytic in the entire complex planeis called an entire function.

Modealg

TheDerivative

AnalyticFunctions

Analytic Functions

A function f is said to be analytic in a domain D if f hasa derivative at every point in D.A function f is analytic at a point z0 if there exists anopen disk B(z0, ε) such that f ′(z) exists for everyz ∈ B(z0, ε).A function f is analytic in a domain D if it is analytic atevery point in D.A function f is analytic in a set S if it is analytic a t everyelement of S.A function which is analytic in the entire complex planeis called an entire function.

Modealg

TheDerivative

AnalyticFunctions

Analytic Functions

A function f is said to be analytic in a domain D if f hasa derivative at every point in D.A function f is analytic at a point z0 if there exists anopen disk B(z0, ε) such that f ′(z) exists for everyz ∈ B(z0, ε).A function f is analytic in a domain D if it is analytic atevery point in D.A function f is analytic in a set S if it is analytic a t everyelement of S.A function which is analytic in the entire complex planeis called an entire function.

Modealg

TheDerivative

AnalyticFunctions

Analytic Functions

A function f is said to be analytic in a domain D if f hasa derivative at every point in D.A function f is analytic at a point z0 if there exists anopen disk B(z0, ε) such that f ′(z) exists for everyz ∈ B(z0, ε).A function f is analytic in a domain D if it is analytic atevery point in D.A function f is analytic in a set S if it is analytic a t everyelement of S.A function which is analytic in the entire complex planeis called an entire function.

Modealg

TheDerivative

AnalyticFunctions

Analytic Functions

A function f is said to be analytic in a domain D if f hasa derivative at every point in D.A function f is analytic at a point z0 if there exists anopen disk B(z0, ε) such that f ′(z) exists for everyz ∈ B(z0, ε).A function f is analytic in a domain D if it is analytic atevery point in D.A function f is analytic in a set S if it is analytic a t everyelement of S.A function which is analytic in the entire complex planeis called an entire function.

Modealg

TheDerivative

AnalyticFunctions

Analytic Functions

A function f is said to be analytic in a domain D if f hasa derivative at every point in D.A function f is analytic at a point z0 if there exists anopen disk B(z0, ε) such that f ′(z) exists for everyz ∈ B(z0, ε).A function f is analytic in a domain D if it is analytic atevery point in D.A function f is analytic in a set S if it is analytic a t everyelement of S.A function which is analytic in the entire complex planeis called an entire function.

Modealg

TheDerivative

AnalyticFunctions

Examples

(a) A polynomial function is analytic at every z ∈ C, sopolynomial functions are entire.

(b) A rational function is analytic at every number in itsdomain.

(c) The function f (z) = |z|2 is nowhere analytic.

Modealg

TheDerivative

AnalyticFunctions

Examples

(a) A polynomial function is analytic at every z ∈ C, sopolynomial functions are entire.

(b) A rational function is analytic at every number in itsdomain.

(c) The function f (z) = |z|2 is nowhere analytic.

Modealg

TheDerivative

AnalyticFunctions

Examples

(a) A polynomial function is analytic at every z ∈ C, sopolynomial functions are entire.

(b) A rational function is analytic at every number in itsdomain.

(c) The function f (z) = |z|2 is nowhere analytic.

Modealg

TheDerivative

AnalyticFunctions

Examples

(a) A polynomial function is analytic at every z ∈ C, sopolynomial functions are entire.

(b) A rational function is analytic at every number in itsdomain.

(c) The function f (z) = |z|2 is nowhere analytic.

Modealg

TheDerivative

AnalyticFunctions

The Cauchy- Riemann Equations

Let f (z) = u(x , y) + iv(x , y) be a function defined in adomain D. If f is analytic in D, then the four partialderivatives ux , uy , vx , vy must exist at every point in D,and these derivatives must satisfy the Cauchy-Riemannequations

ux = vy , uy = vx

In this case, f ′(z) = ux + ivx , or f ′(z) = vy − iuy .Remark: The Cauchy-Riemann equations are anecessary, but not sufficient, condition for a functionf = u + iv to be analytic in a domain D.

Modealg

TheDerivative

AnalyticFunctions

The Cauchy- Riemann Equations

Let f (z) = u(x , y) + iv(x , y) be a function defined in adomain D. If f is analytic in D, then the four partialderivatives ux , uy , vx , vy must exist at every point in D,and these derivatives must satisfy the Cauchy-Riemannequations

ux = vy , uy = vx

In this case, f ′(z) = ux + ivx , or f ′(z) = vy − iuy .Remark: The Cauchy-Riemann equations are anecessary, but not sufficient, condition for a functionf = u + iv to be analytic in a domain D.

Modealg

TheDerivative

AnalyticFunctions

The Cauchy- Riemann Equations

Let f (z) = u(x , y) + iv(x , y) be a function defined in adomain D. If f is analytic in D, then the four partialderivatives ux , uy , vx , vy must exist at every point in D,and these derivatives must satisfy the Cauchy-Riemannequations

ux = vy , uy = vx

In this case, f ′(z) = ux + ivx , or f ′(z) = vy − iuy .Remark: The Cauchy-Riemann equations are anecessary, but not sufficient, condition for a functionf = u + iv to be analytic in a domain D.

Modealg

TheDerivative

AnalyticFunctions

Examples

(a) The function f (z) = z2 is analytic everywhere, so its realand imaginary parts must satisfy the Cauchy-Riemannequations for all points (x , y).

(b) Consider the function defined by

f (z) =

{0 if z = 0

u + iv if z 6= 0

where the functions u and v are defined by

u(x , y) =x3 − y3

x2 + y2 , v(x , y) =x+y3

x2 + y2

It can be shown that the CRE are satisfied at the origin(0, 0) but f ′(0, 0) fails to exist.

(c) Show that the function defined by f (z) = x2 + iy2 isnowhere analytic.

(d) Show: If f is analytic in a domain D and Re f isconstant, then f must be constant in D.

Modealg

TheDerivative

AnalyticFunctions

Examples

(a) The function f (z) = z2 is analytic everywhere, so its realand imaginary parts must satisfy the Cauchy-Riemannequations for all points (x , y).

(b) Consider the function defined by

f (z) =

{0 if z = 0

u + iv if z 6= 0

where the functions u and v are defined by

u(x , y) =x3 − y3

x2 + y2 , v(x , y) =x+y3

x2 + y2

It can be shown that the CRE are satisfied at the origin(0, 0) but f ′(0, 0) fails to exist.

(c) Show that the function defined by f (z) = x2 + iy2 isnowhere analytic.

(d) Show: If f is analytic in a domain D and Re f isconstant, then f must be constant in D.

Modealg

TheDerivative

AnalyticFunctions

Examples

(a) The function f (z) = z2 is analytic everywhere, so its realand imaginary parts must satisfy the Cauchy-Riemannequations for all points (x , y).

(b) Consider the function defined by

f (z) =

{0 if z = 0

u + iv if z 6= 0

where the functions u and v are defined by

u(x , y) =x3 − y3

x2 + y2 , v(x , y) =x+y3

x2 + y2

It can be shown that the CRE are satisfied at the origin(0, 0) but f ′(0, 0) fails to exist.

(c) Show that the function defined by f (z) = x2 + iy2 isnowhere analytic.

(d) Show: If f is analytic in a domain D and Re f isconstant, then f must be constant in D.

Modealg

TheDerivative

AnalyticFunctions

Examples

(a) The function f (z) = z2 is analytic everywhere, so its realand imaginary parts must satisfy the Cauchy-Riemannequations for all points (x , y).

(b) Consider the function defined by

f (z) =

{0 if z = 0

u + iv if z 6= 0

where the functions u and v are defined by

u(x , y) =x3 − y3

x2 + y2 , v(x , y) =x+y3

x2 + y2

It can be shown that the CRE are satisfied at the origin(0, 0) but f ′(0, 0) fails to exist.

(c) Show that the function defined by f (z) = x2 + iy2 isnowhere analytic.

(d) Show: If f is analytic in a domain D and Re f isconstant, then f must be constant in D.

Modealg

TheDerivative

AnalyticFunctions

Examples

(a) The function f (z) = z2 is analytic everywhere, so its realand imaginary parts must satisfy the Cauchy-Riemannequations for all points (x , y).

(b) Consider the function defined by

f (z) =

{0 if z = 0

u + iv if z 6= 0

where the functions u and v are defined by

u(x , y) =x3 − y3

x2 + y2 , v(x , y) =x+y3

x2 + y2

It can be shown that the CRE are satisfied at the origin(0, 0) but f ′(0, 0) fails to exist.

(c) Show that the function defined by f (z) = x2 + iy2 isnowhere analytic.

(d) Show: If f is analytic in a domain D and Re f isconstant, then f must be constant in D.

Modealg

TheDerivative

AnalyticFunctions

Singular Points

Let z0 ∈ C.z0 is called a singular point or a singularity of f if f ′(z0)fails to exist.If z0 is a singularity of f , then z0 is an isolated singularpoint or an isolated singularity of f if there exists ε > 0such that f is analytic everywhere in the punctured disk{z ∈ C : 0 < |z − z0| < ε }.

Modealg

TheDerivative

AnalyticFunctions

Singular Points

Let z0 ∈ C.z0 is called a singular point or a singularity of f if f ′(z0)fails to exist.If z0 is a singularity of f , then z0 is an isolated singularpoint or an isolated singularity of f if there exists ε > 0such that f is analytic everywhere in the punctured disk{z ∈ C : 0 < |z − z0| < ε }.

Modealg

TheDerivative

AnalyticFunctions

Singular Points

Let z0 ∈ C.z0 is called a singular point or a singularity of f if f ′(z0)fails to exist.If z0 is a singularity of f , then z0 is an isolated singularpoint or an isolated singularity of f if there exists ε > 0such that f is analytic everywhere in the punctured disk{z ∈ C : 0 < |z − z0| < ε }.

Modealg

TheDerivative

AnalyticFunctions

Examples

(a) The function f (z) =z3 + 3z

(z + πi)(z2 + 9)is a rational

function, and is hence analytic everywhere in C exceptat z = πi , 3i , −3i .

(b) The function f (z) = Arg z is discontinuous on thenegative ray of the real axis. Thus, f ′(z) fails to exist ateach complex number on this ray.

Modealg

TheDerivative

AnalyticFunctions

Examples

(a) The function f (z) =z3 + 3z

(z + πi)(z2 + 9)is a rational

function, and is hence analytic everywhere in C exceptat z = πi , 3i , −3i .

(b) The function f (z) = Arg z is discontinuous on thenegative ray of the real axis. Thus, f ′(z) fails to exist ateach complex number on this ray.

Modealg

TheDerivative

AnalyticFunctions

Examples

(a) The function f (z) =z3 + 3z

(z + πi)(z2 + 9)is a rational

function, and is hence analytic everywhere in C exceptat z = πi , 3i , −3i .

(b) The function f (z) = Arg z is discontinuous on thenegative ray of the real axis. Thus, f ′(z) fails to exist ateach complex number on this ray.

Modealg

TheDerivative

AnalyticFunctions

Practice Exercises:

(a) Let f be the function defined by

f (z) =

{x4/3y5/3+ix5/3y4/3

x2+y2 z 6= 00 z = 0

Show that f satisfies the Cauchy-Riemann equations at(0, 0), but f ′(0, 0) does not exist.

(b) Suppose that f is analytic in a domain D and thatf ′(z) = 0 for every z ∈ D. Show that f is a constantfunction in D.

(c) Suppose f and g are two functions which are bothanalytic in a domain D, such that f ′(z) = g′(z) for allz ∈ D. Show that f and g differ by an additive constant.

(d) Let f = u + iv be analytic in a domain D, and letg = |f (z)|. Prove that(

∂g∂x

)2

+

(∂g∂y

)2

= |f ′(z)|2

for every z ∈ D.

Modealg

TheDerivative

AnalyticFunctions

Harmonic Functions

Definition: Let u be a real-valued function in a domainD. If all the second-order partial derivatives of u arecontinuous in D and satisfy the Laplace equation:

uxx + uyy =∂2u∂x2 +

∂2u∂y2 = 0

then u is called a harmonic function in D.Example: Show that the function u(x , y) = ex cos y isharmonic for all x , y ∈ R.Theorem: If f = u + iv is analytic in a domain D, then uand v are both harmonic in D.

Modealg

TheDerivative

AnalyticFunctions

Harmonic Functions

Definition: Let u be a real-valued function in a domainD. If all the second-order partial derivatives of u arecontinuous in D and satisfy the Laplace equation:

uxx + uyy =∂2u∂x2 +

∂2u∂y2 = 0

then u is called a harmonic function in D.Example: Show that the function u(x , y) = ex cos y isharmonic for all x , y ∈ R.Theorem: If f = u + iv is analytic in a domain D, then uand v are both harmonic in D.

Modealg

TheDerivative

AnalyticFunctions

Harmonic Functions

Definition: Let u be a real-valued function in a domainD. If all the second-order partial derivatives of u arecontinuous in D and satisfy the Laplace equation:

uxx + uyy =∂2u∂x2 +

∂2u∂y2 = 0

then u is called a harmonic function in D.Example: Show that the function u(x , y) = ex cos y isharmonic for all x , y ∈ R.Theorem: If f = u + iv is analytic in a domain D, then uand v are both harmonic in D.

Modealg

TheDerivative

AnalyticFunctions

Harmonic Functions

Definition: Let u be a real-valued function in a domainD. If all the second-order partial derivatives of u arecontinuous in D and satisfy the Laplace equation:

uxx + uyy =∂2u∂x2 +

∂2u∂y2 = 0

then u is called a harmonic function in D.Example: Show that the function u(x , y) = ex cos y isharmonic for all x , y ∈ R.Theorem: If f = u + iv is analytic in a domain D, then uand v are both harmonic in D.

Modealg

TheDerivative

AnalyticFunctions

Harmonic Conjugate

Definition: Let u and v be harmonic functions in adomain D. Then if the function f = u + iv is analytic inD, then v is called a harmonic conjugate of u.Show that v = 2xy is a harmonic conjugate ofu = x2 − y2 in the entire complex plane.Show that any two harmonic conjugates of the sameharmonic function u differ by an additive constant.If v is a harmonic conjugate of u, is u also a harmonicconjugate of v .Theorem: A function f (z) = u(x , y) + iv(x , y) is analyticin a domain D if and only ifv is a harmonic conjugate ofu.

Modealg

TheDerivative

AnalyticFunctions

Harmonic Conjugate

Definition: Let u and v be harmonic functions in adomain D. Then if the function f = u + iv is analytic inD, then v is called a harmonic conjugate of u.Show that v = 2xy is a harmonic conjugate ofu = x2 − y2 in the entire complex plane.Show that any two harmonic conjugates of the sameharmonic function u differ by an additive constant.If v is a harmonic conjugate of u, is u also a harmonicconjugate of v .Theorem: A function f (z) = u(x , y) + iv(x , y) is analyticin a domain D if and only ifv is a harmonic conjugate ofu.

Modealg

TheDerivative

AnalyticFunctions

Harmonic Conjugate

Definition: Let u and v be harmonic functions in adomain D. Then if the function f = u + iv is analytic inD, then v is called a harmonic conjugate of u.Show that v = 2xy is a harmonic conjugate ofu = x2 − y2 in the entire complex plane.Show that any two harmonic conjugates of the sameharmonic function u differ by an additive constant.If v is a harmonic conjugate of u, is u also a harmonicconjugate of v .Theorem: A function f (z) = u(x , y) + iv(x , y) is analyticin a domain D if and only ifv is a harmonic conjugate ofu.

Modealg

TheDerivative

AnalyticFunctions

Harmonic Conjugate

Definition: Let u and v be harmonic functions in adomain D. Then if the function f = u + iv is analytic inD, then v is called a harmonic conjugate of u.Show that v = 2xy is a harmonic conjugate ofu = x2 − y2 in the entire complex plane.Show that any two harmonic conjugates of the sameharmonic function u differ by an additive constant.If v is a harmonic conjugate of u, is u also a harmonicconjugate of v .Theorem: A function f (z) = u(x , y) + iv(x , y) is analyticin a domain D if and only ifv is a harmonic conjugate ofu.

Modealg

TheDerivative

AnalyticFunctions

Harmonic Conjugate

Definition: Let u and v be harmonic functions in adomain D. Then if the function f = u + iv is analytic inD, then v is called a harmonic conjugate of u.Show that v = 2xy is a harmonic conjugate ofu = x2 − y2 in the entire complex plane.Show that any two harmonic conjugates of the sameharmonic function u differ by an additive constant.If v is a harmonic conjugate of u, is u also a harmonicconjugate of v .Theorem: A function f (z) = u(x , y) + iv(x , y) is analyticin a domain D if and only ifv is a harmonic conjugate ofu.

Modealg

TheDerivative

AnalyticFunctions

Harmonic Conjugate

Definition: Let u and v be harmonic functions in adomain D. Then if the function f = u + iv is analytic inD, then v is called a harmonic conjugate of u.Show that v = 2xy is a harmonic conjugate ofu = x2 − y2 in the entire complex plane.Show that any two harmonic conjugates of the sameharmonic function u differ by an additive constant.If v is a harmonic conjugate of u, is u also a harmonicconjugate of v .Theorem: A function f (z) = u(x , y) + iv(x , y) is analyticin a domain D if and only ifv is a harmonic conjugate ofu.

Modealg

TheDerivative

AnalyticFunctions

Modealg

TheDerivative

AnalyticFunctions

Modealg

TheDerivative

AnalyticFunctions

Modealg

TheDerivative

AnalyticFunctions