Time for some more contour integrals. We have seen earlier examples of contour line integrals and now we look at a more detailed example:

We
want to integrate around the closed contour for which

**f(z) = 3x + 2iy:**
We
check first to see if the function is analytic using the Cauchy –Riemann relations:

Recall
that we require: (see e.g. http://brane-space.blogspot.com/2013/10/looking-at-harmonic-conjugates-and.html )

i)
¶ u/ ¶x = ¶ v/ ¶y and ii) ¶ u/ ¶y
= - ¶ v/ ¶x

For
f(z) = 3x + 2iy = u(x,y) + iv(x,y)

We
have: ¶ u/ ¶x = 3 and
¶ v/ ¶y = 2,
so ¶ u/ ¶x ¹ ¶ v/ ¶y

And:

¶ u/ ¶y = 0 =
- ¶ v/ ¶x

Since
condition (i) is not fulfilled the function is not analytic, hence we cannot
evaluate using Cauchy’ theorem , i.e.

**ò**_{C}^{ }f(z) dz = 0
So
we must integrate line segment by line segment, viz.

**I =**

**ò**

_{C}

^{ }f(z) dz =

**ò**

_{C}^{ }(u + iv) (dx + idy) =

**ò**

_{C}^{ }(3x + 2iy) (dx + idy)

=

**ò**_{C}^{ }(3x dx - 2y dy) + i**ò**_{C}^{ }( 2y dx + 3x dy)
I
= å

^{3 }_{n = 1}_{ }[**ò**_{C}^{ }(3x dx - 2y dy) + i**ò**_{C}^{ }( 2y dx + 3x dy)**On C1: 0**

__<__x__<__1, y = 3, dy = 0**ò**

_{C1}^{ }(3x dx - 2y dy) + i

**ò**

_{C1}^{ }( 2y dx + 3x dy)

= ∫

_{0}^{1 }3x dx + i∫_{0}^{1 }6 dy = [3/2 x^{2}]_{0}^{1 }+ i[6y]_{ 0}^{1 }
= 3/2 + 6i

On
C2:

**3**__<__y__<__5, x = 1, and dx = 0**ò**

_{C2}^{ }(3x dx - 2y dy) + i

**ò**

_{C2}^{ }( 2y dx + 3x dy)

=

**ò**_{3}^{5}^{ }( -2y ) dy + i**ò**_{3}^{5}^{ }3 dy = [- y^{2}]_{3}^{5 }+ i[3y]_{ 3}^{5}
=
- 16 + 6i

On
C3:

**1**__<__t__<__0, x = t, and dx = dt, y = 2t +3, dy = 2dt**Then:**

**ò**

_{C3}^{ }(3x dx - 2y dy) + i

**ò**

_{C3}^{ }( 2y dx + 3x dy)

= ∫

_{1}^{0}^{ }[ 3t dt - 2(2t + 3) 2 dt] + i ∫_{1}^{0}^{ }[ 2(2t + 3) dt + 3t (2dt)]
=
- ∫

_{1}^{0}^{ }(5t + 12) dt + i∫_{1}^{0}^{ }(10t + 6) dt
= - [5t

^{2 }/ 2 + 12t]**+ i[5t**_{ 1}^{0}^{2 }+ 6t]**= 29/2 – 11i**_{ 1}^{0 }
Then
the contour integral value is:

I
= ( 3/2 + 6i) + (-16 + 6i) + (29/2 – 11i) = 0 + i

__Problems for Math Mavens__:

1)
For the closed path shown in the diagram below, evaluate the contour integral. I.

_{}

^{}

Let
f(z) = z

^{2}
f(z)
= 3x + i3y?

## No comments:

Post a Comment