- #1
randybryan
- 52
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I have to integrate |z|2dz from 0 to 1 + 2i using the indicated paths. The first path is a straight line from the origin to 1 + 2i and the second has two lines, the first going from 0 to 2i along the y-axis and then from 2i to 1 + 2i, a line parallel to the x axis.
For the first path, the straight line, I used the parameters z = t +2ti and dz = (1 +2i)dt
|z|2 = 5t2 so the integral became [tex]\int 5t^2 (1+ 2i) dt[/tex] between t=0 and t=1
The answer to this integral is 5/3 (1 + 2i), which is correct according to the back of the book.
However I can't get the line integral along the broken paths to generate the same answer. Can anyone spot an obvious mistake?
For the first path, x= 0 so z = 2it and dz = 2idt
|z|2 = 4t2 so the integral becomes [tex]\int 8it^2dt[/tex] between t = 1 and t=0, giving 8/3 i
For path 2, y = 2i, so dy =0 and it just varies along x.
let z = t + 2i dz= 1 and vary from t= 0 to t= 1
|z|2 = t2 + 4
so the integral becomes [tex]\int (t^2 + 4)dt[/tex] between 0 and 1 and the answer is t2/3 + 4t, between t=0 and t=1 which gives 13/3
now 13/3 and 8/3i does not equal 5/3 (1 +2i)
Where have I gone wrong? And I'm sure I've done something embarrassingly stupid
For the first path, the straight line, I used the parameters z = t +2ti and dz = (1 +2i)dt
|z|2 = 5t2 so the integral became [tex]\int 5t^2 (1+ 2i) dt[/tex] between t=0 and t=1
The answer to this integral is 5/3 (1 + 2i), which is correct according to the back of the book.
However I can't get the line integral along the broken paths to generate the same answer. Can anyone spot an obvious mistake?
For the first path, x= 0 so z = 2it and dz = 2idt
|z|2 = 4t2 so the integral becomes [tex]\int 8it^2dt[/tex] between t = 1 and t=0, giving 8/3 i
For path 2, y = 2i, so dy =0 and it just varies along x.
let z = t + 2i dz= 1 and vary from t= 0 to t= 1
|z|2 = t2 + 4
so the integral becomes [tex]\int (t^2 + 4)dt[/tex] between 0 and 1 and the answer is t2/3 + 4t, between t=0 and t=1 which gives 13/3
now 13/3 and 8/3i does not equal 5/3 (1 +2i)
Where have I gone wrong? And I'm sure I've done something embarrassingly stupid