More Difficult Contour Integrals

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 =     å³ _{n = 1}     [ò  _C  (3x dx  -   2y dy)   + iò  _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 x²] 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)   

=   ò₃ ⁵   ( -2y ) dy  +   i ò₃ ⁵     3 dy  =      [- y²] 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  

=   -  [5t² / 2 + 12t] ₁ ⁰  + i[5t²  + 6t] ₁ ⁰     =  29/2 – 11i

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

   2) For the example shown in the blog post (Fig. 1) , what would the value of I be if:

f(z) = 3x + i3y?