Complex Integrals: Sketching Paths & Computing Integrals

[Maybe_Memorie]

Homework Statement

Sketch the C¹ paths a: [0; 1] -> C, t -> t + it² and b: [0; 1 + i]. Then compute the following integrals.

∫Re(z)dz over a

∫Re(z)dz over b

Homework Equations

The Attempt at a Solution

Sketching a seems ok, y-axis is Imaginary, x-axis is Real, and the path is a quadratic between 0 and 1.

However I'm not sure about b...

As for the integrals, is it just a case of integrating Re(a(t))a'(t)dt between 0 and 1, then integrating Re(a(t))a'(t)dt between 0 and 1+i?

 

[I like Serena]

Hi MM!

I believe you've got (a) down.

For (b) I have to admit I haven't seen the notation [0; 1 + i] before.
Do you have a definition for it?
I'd assume it's supposed to represent the line segment between 0 and (1+i).
Or in other words: (1+i)[0, 1].
It looks a bit weird though, since a curve is usually defined on an interval of real numbers.
However, if this is the case, you need to reconsider what Re(z) is.

 

[Maybe_Memorie]

Hi MM!

I believe you've got (a) down.

For (b) I have to admit I haven't seen the notation [0; 1 + i] before.
Do you have a definition for it?
I'd assume it's supposed to represent the line segment between 0 and (1+i).
Or in other words: (1+i)[0, 1].
It looks a bit weird though, since a curve is usually defined on an interval of real numbers.
However, if this is the case, you need to reconsider what Re(z) is.

I don't have a definition for it. I'll upload the actual problem to show it to you.

 

[I like Serena]

I don't have a definition for it. I'll upload the actual problem to show it to you.

I get it, in particular if I look at the problems that are coming, that show which theory you're currently learning.

Path b is not related to path a.
It is just a (linear) path from 0 to (1+i).
It would be given by b:[0,1]→C defined by t→t(1+i)

Note that path a also starts in 0 and ends in (1+i).

And also note that for path a the interval is [0,1] and not [0;1].
[0,1] is a real interval, whereas [0;1] would indicate a (linear) path between 0 and 1 in the complex plane.