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Cauchy’s Theorems I

Ang M.S.

Augustin-Louis Cauchy
1789 – 1857

October 26, 2012

References

Murray R. Spiegel Complex V ariables with introduction to conformal mapping and its applications Dennis G. Zill , P. D. Shanahan A First Course in Complex Analysis with Applications J. H. Mathews , R. W. Howell Complex Analysis For Mathematics and Engineerng

1 Summary

• Cauchy Integral Theorem

Let f be analytic in a simply connected domain D. If C is a simple closed contour that lies in D , and there is no singular point inside the contour, then

ˆ

f (z) dz = 0

C

• Cauchy Integral Formula (For simple pole)

If there is a singular point z0 inside the contour, then

˛

f(z)

dz = 2πj f(z0) z − z0

• Generalized Cauchy Integral Formula (For pole with any order)

˛

f(z)

(z − z0)n
2πj

  • dz =
  • f

(n−1) (z0)

(n − 1)!

• Cauchy Inequality

  • ꢀ
  • ꢀ

ꢀ

M · n!

(n)

ꢀ

f

(z0) ≤

rn

• Gauss Mean Value Theorem

ˆ

2π

  • (
  • )

1f(z0) =

f z0 + rejθ dθ
2π

0

1

2 Related Mathematics Review

2.1 Stoke’s Theorem

  • ¨
  • ˛

  • ¯
  • ¯

  • ∇ × F · dS =
  • F · dr

  • Σ
  • ∂Σ

(The proof is skipped)

¯

Consider F = (FX , FY , FZ)

xˆ

∂

yˆ

∂

zˆ

∂



¯



∇ × F = det

∂x ∂y ∂z

FX FY FZ

Let FZ = 0

xˆ

∂

yˆ

∂

zˆ

∂




¯
∇ × F = det

∂x ∂y ∂z

FX FY

0
¯

dS = nˆdS , for dS = dxdy , nˆ = zˆ . By zˆ · zˆ = 1 , consider zˆ component only for ∇ × F

xˆ

∂

  • yˆ
  • zˆ

∂

  • (
  • )




  • ∂
  • ∂FY

∂x
∂FX
∂y

¯
∇ × F = det
=

−

zˆ

∂x ∂y ∂z

FX FY

0

i.e.

  • (
  • )

  • ¨
  • ¨

∂FY
∂x
∂FX
∂y

¯

∇ × F · dS =

−

dxdy

  • Σ
  • Σ

Consider the RHS (let FZ = 0)

  • ˛
  • ˛
  • ˛

¯

F · dr =

  • (FX, FY , FZ) · (dx, dy, dz) =
  • FXdx + FY dy

  • ∂Σ
  • ∂Σ
  • ∂Σ

Thus

  • (
  • )

  • ¨
  • ˛

∂FY
∂x
∂FX
∂y

−

  • dxdy =
  • FXdx + FY dy

  • Σ
  • ∂Σ

2.2 Cauchy-Reimann Condition

For f(z) = u(z) + jv(z) = u(x, y) + jv(x, y)

The complex derivative is

f (z0 + h) − f(z0)

  • lim
  • = f ′ (z0)

hh → 0

h ∈ C

If the limit exists.
2
If the limit exists, limits along real axis should be the same the limit along imaginary axis

f (z0 + h) − f (z0)

∂f (z0)

1 ∂f (z0)

f (z0 + jh) − f (z0)

  • lim
  • =
  • =
  • =
  • lim

  • h
  • ∂x
  • j
  • ∂y
  • jh

  • h → 0
  • h → 0

  • h ∈ R
  • h ∈ R

i.e. The Cauchy-Riemann Equation

∂f (z0)

1 ∂f (z0)
=

  • ∂x
  • j
  • ∂y

Expand

∂

1 ∂

  • [u (x0, y0) + jv (x0, y0)] =
  • [u (x0, y0) + jv (x0, y0)]

  • ∂x
  • j ∂y

Equalize real part and imaginary part

  • ∂
  • ∂



  • u (x0, y0) =
  • v (x0, y0)

∂x ∂
∂y
∂



v (x0, y0) = − u (x0, y0)

  • ∂x
  • ∂y

Or simply as

∂u ∂x
∂v ∂y



=

  • ∂v
  • ∂u



= −

  • ∂x
  • ∂y

i.e. If f(z) is analytic, then it fulfill this condition

2.3 ML Inequality

First,

  • ˆ
  • ˆ

  • b
  • b

if g(x) ≤ f(x) , then

  • g(x)dx ≤
  • f(x)dx

  • a
  • a

This is true if

  • sup g(x) ≤ sup f(x)
  • inf g(x) ≤ inf f(x)

Then for any function f , it is true that

−|f| ≤ f ≤ |f|

Then apply the inequality above ,

  • ˆ
  • ˆ
  • ˆ

b

−

  • b |f(x)|dx ≤
  • f(x)dx ≤ b |f(x)|dx

  • a
  • a
  • a

Now is the time to show , for bounded f(z) , i.e. |f(z)| ≤ M ,

  • ꢀ
  • ꢀ

ˆ

ꢀꢀꢀ
ꢀꢀ

f(z)dz ≤ ML

ꢀ

c

3
Pf.

  • ´
  • ´

b

By using the b f(x)dx ≤ |f(x)|dx

  • a
  • a

ꢀ

ˆ

ꢀ

ˆ

ꢀꢀꢀ
ꢀꢀ

f(z)dz ≤ |f(z)|dz

ꢀ

  • c
  • c

Since f(z) si bounded,

  • ꢀ
  • ꢀ

  • ˆ
  • ˆ
  • ˆ
  • ˆ

ꢀꢀꢀ
ꢀꢀ

  • f(z)dz ≤ |f(z)|dz ≤ Mdz = M
  • dz = ML

ꢀ

  • c
  • c
  • c
  • C

| {z }

L

Therefore

ˆ

f(z)dz| ≤ ML

  • |
  • |

c

3 The Cauchy Theorem

f(z) = u(x, y) + jv(x, y)

  • ˛
  • ˛
  • ˛
  • ˛

  • f (z) dz =
  • [u (x, y) + jv (x, y)] [dx + jdv] =
  • udx − vdy + j
  • vdx + udy

  • C
  • C
  • C
  • C

3.1 The Real Part

Let FX = u (x, y) FY = −u (x, y) and consider the real part of the integral

  • ˛
  • ˛

udx − vdy = FXdx + FY dy

C

By the Stoke’ Theorem

  • ˛
  • ¨
  • ¨

udx + (−v)dy =

[(−v)x − uy] dxdy =

[−vx − uy] dxdy

C

Since the function is analytic in the region D , ⇐⇒ the function fulfill Cauchy-Reimann Condition

∂v ∂x
∂u ∂y

= −

Then

  • ¨
  • ¨

[−vx − uy] dxdy =

[uy − uy] dxdy = 0

´

The real part of C f(z)dz equal zero
4

3.2 The imaginary part

´

Consider the imaginary part of C f(z)dz ,

˛

vdx + udy

C

Let FX be v(x, y) and FY be u(x, y) , and apply Stoke’s Theorem

  • ˛
  • ¨

  • vdx + udy =
  • (ux − vy) dxdy

C

∂u ∂x
∂v ∂y

As the function is analytic, so it fulfill Cauchy-Reimann Condition : become zero

=

, thus the integral
Finally, Let f be analytic in a simply connected domain D , for the simple closed contour C that lies in D , the close contour integral equal zero

ˆ

f (z) dz = 0

C

4 The Cauchy Integral Formulas

4.1 Simple Pole

˛

g (z) dz = 2πjg (z0) z − z0

C

Condition • z0 inside C , if z0is outside C , then the integral is just zero • g(z) is analytic in simply connected domain ⇐⇒ domain of g(z) is simply connect closed region, and g(z) fulfill CR-Condition there

Proof. Consider the diagram

g(z)

, the integral of

By concept of path deformation z − z0

along path C that ϵ → 0 along path C is equivalent to the integral

ϵ

  • ˛
  • ˛

  • g(z)
  • g(z)

dz = lim

dz

ϵ→0

  • z − z0
  • z − z0

C

Cϵ

The small circle inside can be expressed as

|z − z0| = ϵejθ 0 ≤ θ < 2π

As the radius ϵ → 0+ , the circle is thus

z − z0 = ϵejθ 0 ≤ θ < 2π

5
Put this back into the integral

  • (
  • )

  • ˛
  • ˛
  • ˆ

g z0 + ϵejθ

(z0 + ϵejθ) − z0

2π

  • (
  • )

  • g(z)
  • g(z)

  • dz = lim
  • dz = lim

d z0 + ϵejθ

  • ϵ→0
  • ϵ→0

  • z − z0
  • z − z0

C

Cϵ

0

  • (
  • )

  • ˆ
  • ˆ

2π

(

g z0 + ϵejθ

2π

)

= lim

  • jϵejθdθ = jlim
  • g z0 + ϵejθ dθ

ϵejθ

  • ϵ→0
  • ϵ→0

  • 0
  • 0

  • ˆ
  • ˆ
  • ˆ

  • [
  • ]

)

  • 2π
  • 2π
  • 2π

(

= j

  • limg z0 + ϵejθ dθ = j
  • g (z0) dθ = jg (z0)
  • dθ = 2πjg (z0)

ϵ→0

  • 0
  • 0
  • 0

∴

˛

g(z)

dz = 2πjg (z0) z − z0

C

4.2 Generalized Cauchy Integral Formula

˛

  • f (z0)
  • 2πj

  • dz =
  • 2πj f(n−1) (z0)

(z − z0)n

(n − 1)!

First, the Cauchy Integral Formula for simple pole is

˛

g (z) dz = 2πjg (z0) z − z0

C

(

on both side Not

˛

)

  • d
  • d

Take

!

  • dz0
  • dz

˛

g (z)

⇒

  • d
  • g (z)
  • d
  • 2πj

  • dz = 2πj
  • g (z0)
  • dz =

g′ (z0)

(z − z0)2

  • dz0
  • z − z0
  • dz0

1!

  • C
  • C

Repeat,

  • ˛
  • ˛

g (z)

⇒

  • d
  • g (z)

(z − z0)2

g (z)
(z − z0)3

  • 2πj d
  • 2πj

dz =

g′ (z0)

dz =

g” (z0)

(z − z0)3 dz0

  • 1! dz0
  • 2!

  • C
  • C

  • ˛
  • ˛

g (z)

⇒

  • d
  • 2πj d
  • 2πj

dz =

g” (z0)

  • dz =
  • g

(3) (z0)

(z − z0)4 dz0

  • 1! dz0
  • 3!

  • C
  • C

Thus, the general form of Cauchy Integral Formula is

˛

  • g (z)
  • 2πj

  • dz =
  • g

(n−1) (z0)

(z − z0)n

(n − 1)!

C

This can be proved using Mathematical Induction.
6

5 Consequences of Cauchy’s Integral Formula

5.1 Cauchy Inequality

The generalized Cauchy Integral Formula is

˛

f (z)

(z − z0)n+1

2πj

  • dz =
  • f

(n) (z0)

n!

C

Take absolute value

  • ꢀ
  • ꢀ

ꢀ
ꢀꢀꢀꢀ
ꢀꢀꢀ

  • ꢀ
  • ꢀ

ꢀ

  • ˛
  • ˛

  • ꢀ
  • ꢀ

ꢀꢀ
ꢀꢀ
ꢀꢀ

f (z)

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  • The Mean Value Theorem and Analytic Functions of a Complex Variable

    The Mean Value Theorem and Analytic Functions of a Complex Variable

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. Math. Anal. Appl. 324 (2006) 30–38 www.elsevier.com/locate/jmaa The mean value theorem and analytic functions of a complex variable M.A. Qazi Department of Mathematics, Tuskegee University, Tuskegee, AL 36088, USA Received 9 August 2005 Available online 10 January 2006 Submitted by A.V. Isaev Abstract The mean value theorem for real-valued differentiable functions defined on an interval is one of the most fundamental results in Analysis. When it comes to complex-valued functions the theorem fails even if the function is differentiable throughout the complex plane. we illustrate this by means of examples and also present three results of a positive nature. © 2005 Elsevier Inc. All rights reserved. Keywords: Mean value theorem; Entire functions; Quadrinomials; Univalent functions We start by recalling the mean value theorem [8, p. 93] of real analysis. Theorem A. Let f be a real continuous function on a closed interval [a,b] which is differentiable in the open interval (a, b). Then there is a point ξ ∈ (a, b) at which f(b)− f(a)= f (ξ)(b − a). This result extends to functions whose domains lie in the complex plane provided their ranges are on the real line.Infact,ifΩ is a convex domain in the n-dimensional Euclidean space, and f is real-valued and differentiable on Ω, then f(b)− f(a)= Df (ξ ) · (b − a) = grad f(ξ),b− a (a ∈ Ω, b ∈ Ω), E-mail address: [email protected].
  • Mean Value Theorem on Manifolds

    Mean Value Theorem on Manifolds

    MEAN VALUE THEOREMS ON MANIFOLDS Lei Ni Abstract We derive several mean value formulae on manifolds, generalizing the clas- sical one for harmonic functions on Euclidean spaces as well as the results of Schoen-Yau, Michael-Simon, etc, on curved Riemannian manifolds. For the heat equation a mean value theorem with respect to `heat spheres' is proved for heat equation with respect to evolving Riemannian metrics via a space-time consideration. Some new monotonicity formulae are derived. As applications of the new local monotonicity formulae, some local regularity theorems concerning Ricci flow are proved. 1. Introduction The mean value theorem for harmonic functions plays an central role in the theory of harmonic functions. In this article we discuss its generalization on manifolds and show how such generalizations lead to various monotonicity formulae. The main focuses of this article are the corresponding results for the parabolic equations, on which there have been many works, including [Fu, Wa, FG, GL1, E1], and the application of the new monotonicity formula to the study of Ricci flow. Let us start with the Watson's mean value formula [Wa] for the heat equation. Let U be a open subset of Rn (or a Riemannian manifold). Assume that u(x; t) is 2 a C solution to the heat equation in a parabolic region UT = U £ (0;T ). For any (x; t) de¯ne the `heat ball' by 8 9 jx¡yj2 < ¡ 4(t¡s) = e ¡n E(x; t; r) := (y; s) js · t; n ¸ r : : (4¼(t ¡ s)) 2 ; Then Z 1 jx ¡ yj2 u(x; t) = n u(y; s) 2 dy ds r E(x;t;r) 4(t ¡ s) for each E(x; t; r) ½ UT .
  • Multivariable Calculus Review

    Multivariable Calculus Review

    Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Multivariable Calculus Review Multivariable Calculus Review Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Multivariable Calculus Review n I ν(x) ≥ 0 8 x 2 R with equality iff x = 0. n I ν(αx) = jαjν(x) 8 x 2 R α 2 R n I ν(x + y) ≤ ν(x) + ν(y) 8 x; y 2 R We usually denote ν(x) by kxk. Norms are convex functions. lp norms 1 Pn p p kxkp := ( i=1 jxi j ) ; 1 ≤ p < 1 kxk1 = maxi=1;:::;n jxi j Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Multi-Variable Calculus Norms: n n A function ν : R ! R is a vector norm on R if Multivariable Calculus Review n I ν(αx) = jαjν(x) 8 x 2 R α 2 R n I ν(x + y) ≤ ν(x) + ν(y) 8 x; y 2 R We usually denote ν(x) by kxk. Norms are convex functions. lp norms 1 Pn p p kxkp := ( i=1 jxi j ) ; 1 ≤ p < 1 kxk1 = maxi=1;:::;n jxi j Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Multi-Variable Calculus Norms: n n A function ν : R ! R is a vector norm on R if n I ν(x) ≥ 0 8 x 2 R with equality iff x = 0.
  • A Mean Value Theorem for Generalized Riemann Derivatives

    A Mean Value Theorem for Generalized Riemann Derivatives

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 2, February 2008, Pages 569–576 S 0002-9939(07)08976-9 Article electronically published on November 6, 2007 A MEAN VALUE THEOREM FOR GENERALIZED RIEMANN DERIVATIVES H. FEJZIC,´ C. FREILING, AND D. RINNE (Communicated by David Preiss) Abstract. Functional differences that lead to generalized Riemann deriva- tives were studied by Ash and Jones in (1987). They gave a partial answer as to when these differences satisfy an analog of the Mean Value Theorem. Here we give a complete classification. 1. Introduction Throughout this article, ∆ will denote the following functional difference: n ∆f(x)= aif(x + bih) , i=1 where a1,...,an and b1 <b2 < ··· <bn are constants. We will also suppose that there is some positive integer d, called the order of ∆, such that n j j ≤ (1.1) ∆ = aibi =0for0 j<d, i=1 n d d (1.2) ∆ = aibi = d! (normalized). i=1 A generalized Riemann derivative is created by ∆f(x) lim h→0+ hd when this limit exists. It is a well-known fact that if the ordinary dth derivative, f (d)(x), exists, then so does the generalized Riemann derivative and the two must agree. In the case where f (d) is continuous at x this can be easily seen by d- applications of l’Hˆopital’s Rule. More generally, it follows from a variant of Taylor’s Theorem first proved by Peano (see pp. 245–246 of [2] for a statement and history). Definition 1.1. We will say that ∆ possesses the mean value property if for any x,anyh>0, and any function f(x) such that f (d−1)(x) is continuous on Received by the editors March 28, 2006 and, in revised form, October 10, 2006.