2 Dec 2011

School

Department

Course

Professor

Integral formulas for area

November 29, 2011

During class, I gave the following formula for calculating the area Ainside

a closed, positively-oriented, contour C:

1

2

Im

I

C

z dz. (1)

The text gives the following formula in Exercise 7, page 163:

1

2i

I

C

z dz. (2)

This seems confusing. Which formula is correct?

Well, I derived Equation (1) in class, and so it must be correct. To see what’s

wrong with Equation (2), let’s split the integral into its real and imaginary parts

I

C

z dz =X+iY,

so that Equation (1) becomes

Area =

1

2

Im

I

C

z dz =

1

2

Y.

Equation (2) on the other hand is

1

2i

I

C

z dz =

1

2i

(X+iY ) =

1

2

Y

|{z}

Area

−

i

2

X,

So it is only the real part of Equation (2) that corresponds to the area enclosed

by the contour C.

1