Cauchy Integral Formula

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Jun 13, 2015 - Cauchy Integral Formula. M Azram. 1 and F A M Elfaki. Department of Science, Faculty of Engineering. IIUM, Kuala Lumpur 50728, Malaysia.
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Cauchy Integral Formula

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2013 IOP Conf. Ser.: Mater. Sci. Eng. 53 012003 (http://iopscience.iop.org/1757-899X/53/1/012003) View the table of contents for this issue, or go to the journal homepage for more

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5th International Conference on Mechatronics (ICOM’13) IOP Conf. Series: Materials Science and Engineering 53 (2013) 012003

IOP Publishing doi:10.1088/1757-899X/53/1/012003

Cauchy Integral Formula M Azram1 and F A M Elfaki Department of Science, Faculty of Engineering IIUM, Kuala Lumpur 50728, Malaysia [email protected]

ABSTRACT: Cauchy-Goursat integral theorem is D pivotal, fundamentally important, and well celebrated result in complex integral calculus. It requires analyticity of the function inside and on the boundary of the simple closed curve. In this study we will investigate the condition(s) under which

³ f ( z )dz c

0 even though the function is not analytic at a point inside C. Consequently,

we will extend the above notion to a finite numbers of points and will present an easy and simple proof of unquestionably the most important, significant and pivotal result known as Cauchy integral formula.

1. INTRODUCTION Complex variable is an extension of real calculus which in fact discloses everything hidden in the real calculus. Complex integration is pivotal in the study of complex variables. As in calculus, the fundamental theorem of calculus is significant because it relates integration with differentiation and at the same time provides method of evaluating integral so is the complex analog to develop integration along arcs and contours is complex integration. Complex integration is elegant, powerful and a useful tool for mathematicians, physicists and engineers. Cauchy-Goursat theorem is pivotal, fundamentally important and well celebrated theorem of the complex integral calculus. This theorem is not only a pivotal result in complex integral calculus but is frequently applied in quantum mechanics, electrical engineering, conformal mappings, method of stationary phase, mathematical physics and many other areas of mathematical sciences and engineering. It provides a convenient tool for evaluation of a wide variety of complex integration. It also forms the cornerstone of the development of results f c(z ) of an analytic function f (z ) is analytic, Cauchy’s integral formula and many advance topics in complex integration. Due to its pivotal role and importance, mathematicians have discussed it in all respects [3-5,7,8].

M Azram, Department of Science, Faculty of Engineering, IIUM, Kuala Lumpur 50728, Malaysia

[email protected]

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1

5th International Conference on Mechatronics (ICOM’13) IOP Conf. Series: Materials Science and Engineering 53 (2013) 012003

IOP Publishing doi:10.1088/1757-899X/53/1/012003

Historically, it was firstly established by A.L. Cauchy (1789-1857) [1] and later on extended by E. Goursat (1858-1936) [1] without assuming the continuity of f c(z ) . Azram [6] has given a variational but simple proof of Cauchy-Goursat integral theorem. Cauchy-Goursat integral theorem has laid down the deeper foundations for Cauchy- Riemann theory of complex variables. The line integral of a complex function is mostly dependent on the endpoints of the path of integration as well as on the choice of the path. This dependence often complicates situations. Hence, condition(s) under which this does not occur are of considerable importance and is due to Cauchy-Goursat integral theorem. Cauchy-Goursat integral theorem leads to a remarkable Cauchy integral formula which shows that the values of an analytic function at the interior points can be determined by the values of the function on the boundary. Another important and useful consequence of Cauchy's integral formula is that every analytic function is infinitely differentiable. Guy Jumarie [4] has studied the Cauchy integral formula via the modified Riemann–Liouville derivative. The following are also important results resulting as a consequence of Cauchy Integral formula; Cauchy’s estimate Cayley-Hamilton theorem, Cauchy’s inequality , Liouville’s theorem, Fundamental theorem of algebra, Gauss’ mean value theorem, Maximum modulus theorem and Minimum modulus theorem etc. Cauchy-Goursat integral theorem requires analyticity of the function inside and on the boundary of the simple closed curve. In this study we will investigate the condition(s) under which ³ f ( z )dz 0 even

c though the function is not analytic at a point inside C. Consequently, we will extend the above notion to a finite numbers of points and will present an easy and simple proof of unquestionably the most important, significant and pivotal result known as Cauchy integral formula. 2. RESULTS AND DISCUSSION Theorem 1. If a function f (z ) is analytic inside and on a simple closed curve point z0 such that

lim f ( z ) 0

z o z0

then ³ f ( z )dz

c

except at an interior

0

c

Proof. Let f (z ) be analytic inside and on a simple closed clockwise oriented curve c except at an interior

point

z0 .

Since

lim f z

z o z0

)0(

Ÿg

i

v H e 0 Gn !0!

f ( z)  H whenever 0  z  z0  G . Now, choose a clockwise oriented circle C1 of radius r centre at z0 that is C1 : z  z0

r . The circle C1 is completely inside C and 0  r  G (Figure 1).

2

5th International Conference on Mechatronics (ICOM’13) IOP Conf. Series: Materials Science and Engineering 53 (2013) 012003

Since

IOP Publishing doi:10.1088/1757-899X/53/1/012003

lim f ( z) 0 Ÿ lim ( z  z0 ) f ( z) 0 Ÿ

z o z0

z o z0

( z  z0 ) f ( z )  H whenever 0  z  z0  G Ÿ f ( z )  By well-known results in complex integration ³ f ( z )dz

c

³ f ( z)dz

³ f ( z)dz d ³

c

c1

f ( z )dz 

c1

H r

2S r

2SH Ÿ

H

H

z  z0

r

³ f ( z )dz Ÿ c1

³ f ( z)dz c

0 . Hance, ³ f ( z )dz c

0.

This

completes the proof. Analogue to the above result, we can extend the result to a finite numbers of points that is; Theorem 2. numbers

If a function f (z ) is analytic inside and on a simple closed curve

of

interior

n ( ) f z dz ¦ ³ f ( z ) dz ³ i 0 ci c

points

zi : i

0, 1,2 3,    n

such

that

c

except at a finite

lim f ( z ) 0

z o zi

then

0

CAUCH INTEGRAL FORMULA Theorem 3. then

f (z )

Let f (z ) be analytic inside and on a simple closed curve C and z0 is a point inside C

³ z  z00 dz

2S i f ( z0 ) .

c

3

5th International Conference on Mechatronics (ICOM’13) IOP Conf. Series: Materials Science and Engineering 53 (2013) 012003

Proof.

Since f (z ) is analytic inside and on C so is at z0 Ÿ f c( z0 )

lim ( z  z0 ) f c( z0 ) z o z0

lim ( z  z0 ) z o z0

f ( z )  f ( z0 ) z  z0

Applying the result of theorem 1, we can get

³ c

³ c

IOP Publishing doi:10.1088/1757-899X/53/1/012003

f ( z )  f ( z0 ) dz z  z0

0Ÿ³ c

f ( z) dz z  z0

f (z )

f ( z0 ) ³

c

c

0 . Now

1 dz z  z0

Now, choose a circle C1 inside C of radius r centre at z0 that is C1 : z  z0

z  z0

r Ÿ z  z0

f ( z )  f ( z0 ) exist. z  z0

0

f ( z )  f ( z0 ) dz z  z0

³ z  z00 dz

lim z o z0

reiT where 0 d T  2S Ÿ dz

(1)

r.

i reiT dT

Consequently result in eq (1) changes as

f ( z) ³ z  z0 dz c

f ( z0 ) ³ c

1 dz z  z0

f ( z0 ) ³ c1

1 dz z  z0

2S f ( z0 )

³

0

i r eiT dT i T re

2S i f ( z0 ) .

This

completes the proof . 3. CONCLUSION Cauchy-Goursat theorem is the basic pivotal theorem of the complex integral calculus. Theorem 1 provides us a kind of extension to Cauchy-Goursat theorem. Significance of theorem 1 is that it provides us a simple proof of Cauchy integral formula avoiding strict and rigor mathematical requirements.

REFERENCES [1]

Churchill R V and James W B 2003 Complex Variables and Applications 7th Ed. McGraw Hill Inc.

[2]

Jumarie G 2010 Cauchy’s integral formula via the modified Riemann–Liouville derivative for analytic functions of fractional order Applied Mathematical Letters, 23(12) 1444-1450

[3]

Gario P 1981 Cauchy’s Theorem on the Rigidity of Convex Polyhrdra Archimede 33(1-2) 5369

[4]

Martins and Luiz C 1976 On Cauchy’s Theorem in Classical Physics, Arch. Rational Mech. Anal. 60(4) 305-324

4

5th International Conference on Mechatronics (ICOM’13) IOP Conf. Series: Materials Science and Engineering 53 (2013) 012003

IOP Publishing doi:10.1088/1757-899X/53/1/012003

[5]

Mibu Y 1959 Cauchy’s Theorem in Banach Spaces J. Math. Soc. Japan 11 76-82

[6]

Azram M, Daoud J I and Elfaki F A M 2010 On the Cauchy-Goursat Theorem J. Applied Sci. 10(13) 1349-135

[7]

Blaya R A, Reyes J B, Brackx F, Schepper H D and Sommen F 2012 Cauchy Integral Formulae in Quaternionic Hermitean Clifford Analysis Complex Analysis and Operator Theory 6(5) 971-985

[8]

Segev R and Rodnay G 2000 Cauchy Theorem’s on Manifolds J. Elasticity 56(2) 129-144

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