java barcode scanner api Complex Variables Demysti ed in Java

Make Denso QR Bar Code in Java Complex Variables Demysti ed

This result can be extended further We consider a domain D in the complex plane and a curve (t ) which maps a real, closed interval a t b into D If there is a continuously differentiable function h, which maps D into the real numbers, then the integral along the curve is given by

h df h dg ) x (t ) dt + y (t ) dt dt = h( (b)) h( (a))

(610)

DEFINITION: COMPLEX LINE INTEGRAL OR CONTOUR INTEGRAL Now suppose that the curve (t ) is a simple closed curve Then the complex line integral of a function F ( z ) of a complex variable is written as

F ( (t ))

(611)

The integral in Eq (611) is known as a contour integral EXAMPLE 64 Suppose that 0 t 1, f ( z ) = z and we integrate along the curve (t ) = 1 + (i 1)t Calculate f ( z )dz SOLUTION This can be done by using Eq (611) Given that (t ) = 1 + (i 1)t , we see that d = i 1 dt We also have that f ( z ) = f ( (t )) = z = 1 + (i 1)t and so

F ( (t ))

t2 1 = (i 1) t + (i 1) 2 0 i + 1 = (i 1) = 1 2

EXAMPLE 65 Suppose that f ( z ) = z 2 + 1 Integrate f ( z ) around the unit circle SOLUTION We can integrate around the unit circle by de ning the curve:

The interval mapped by this curve to the complex plane is 0 t 2 We nd that the derivative of the curve is d d = (eit ) = ieit dt dt Using Eq (611) we have

f ( z ) dz = f ( (t ))

a 2 0

0 2

2 1 = ei 3 t + eit 3 0 1 = (ei 6 1) + (ei 2 1) = 0 3 This result follows since for any even m, eim = cos(m ) + i sin(m ) = 1 + i 0 = 1

Now let s take a turn that we re going to use to develop the groundwork for residue theory, the topic of the next chapter First let s begin by looking at complex integration once again We ll dispense with the parameter t and instead focus on functions of x and y So we have w = f ( z ) = u( x , y) + iv ( x , y)

(612)

So we can write the integral of a complex function along a curve in the following way:

(613)

With this in hand, we can de ne the fundamental theorem of calculus for a function of a complex variable as follows Suppose that f ( z ) has an antiderivative That is: f (z) = dF dz

f (z ) dz = dz dz = F (z ) a = F (b) F (a)

(614)

To prove this result, we use F ( z ) = U + iV We are assuming that f and F are analytic Now, using the results of Chap 3 we know that f (z) = and so U dF U V V i = = +i y dz x x y

= F (b) F (a) The fundamental theorem of calculus allows us to evaluate many integrals in the usual way EXAMPLE 66

n Find f ( z )dz when f ( z ) = z for the case of z (= a) z (= b), and when the curve is closed, that is when a = b

SOLUTION The fundamental theorem allows us to evaluate the integral in the same way we would in the calculus of real variables We have