Contour integration by parametisation

[Poirot]

Homework Statement

Determine, by explicit parameterisation,

∫_C dz f(z), where f(z) = z^n , n ∈ Z, n > 0
and C is the line segment from z = 0 to z = 2 together with the line segment from z = 2 to z = 2 + i.

Homework Equations

∫_ϒ f(z) dz = ∫_a^b dt f(ϒ(t))ϒ'(t) where ϒ'(t)≈dz/dt

The Attempt at a Solution

Splitting C into two sub contours ϒ₁ = {z=x, x ∈ [0 , 2]} and ϒ₂ = {z=y, y ∈ [2 , 2+i]}. Then using the equation above to integrate over both and sum. This gave and answer which I really can't gauge whether is correct or not.

(2+i)⁽ⁿ⁺¹⁾/(n+1)

But my lecturer gave us a rough check guide and said that if what you start with is a real number then your answer should be real also. Thus am skeptical of my answer.

Any help on the method, answer or general help with this would be greatly appreciated.

Thanks!

 

[SteamKing]

Homework Statement

Determine, by explicit parameterisation,

∫_C dz f(z), where f(z) = z^n , n ∈ Z, n > 0
and C is the line segment from z = 0 to z = 2 together with the line segment from z = 2 to z = 2 + i.

Homework Equations

∫_ϒ f(z) dz = ∫_a^b dt f(ϒ(t))ϒ'(t) where ϒ'(t)≈dz/dt

The Attempt at a Solution

Splitting C into two sub contours ϒ₁ = {z=x, x ∈ [0 , 2]} and ϒ₂ = {z=y, y ∈ [2 , 2+i]}. Then using the equation above to integrate over both and sum. This gave and answer which I really can't gauge whether is correct or not.

(2+i)⁽ⁿ⁺¹⁾/(n+1)

But my lecturer gave us a rough check guide and said that if what you start with is a real number then your answer should be real also. Thus am skeptical of my answer.

Any help on the method, answer or general help with this would be greatly appreciated.

Thanks!

It's not clear what your lecturer was talking about, or if you have fully understood what he said, but I have never heard of such a "rough check" on open contour integrals.

In fact, a similar problem is contained in this article (pp. 4-5) where f(z) = z² and the contour is dog-legged like the one for your problem:

https://math.dartmouth.edu/archive/m43s12/public_html/notes/class6.pdf

The calculation starts out with a real result on C1 and gives a complex result on C2 with a net result which is complex.