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Explain solution please

nacho

Active member
Sep 10, 2013
156
Referring to the attached image.

i seem to have forgotten this material and am trying to revise,

how/why does it become $\frac{1}{2}$ $(\sin(z))^2 $' ?

and does the path $C$ that we are given come into play in the solution?

Sorry for the noobness, any help is appreciated!
 

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Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,403
It doesn't. [tex]\displaystyle \begin{align*} \sin{(z)}\cos{(z)} = \frac{1}{2}\sin{(2z)} \end{align*}[/tex], not [tex]\displaystyle \begin{align*} \frac{1}{2}\sin^2{(z)} \end{align*}[/tex]...
 

nacho

Active member
Sep 10, 2013
156
I should clarify I meant how did they go from question to step 1, then step 2, step 3.

what is being done?
 

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,403
Couldn't be more wrong I'm afraid. First of all, this is a LINE INTEGRAL. The so-called "Solution" has not taken that into account at all. In fact, there has not been any integrating done at all either...
 

chisigma

Well-known member
Feb 13, 2012
1,704
Referring to the attached image.

i seem to have forgotten this material and am trying to revise,

how/why does it become $\frac{1}{2}$ $(\sin(z))^2 $' ?

and does the path $C$ that we are given come into play in the solution?

Sorry for the noobness, any help is appreciated!
The function $\displaystyle f(z) = \frac{1}{2}\ \sin 2 z $ is analytic over rhe entire complex plane, so that the line integral is independent from the path connecting the point 0 to the point 1 + i. That means that is...

$\displaystyle \frac{1}{2} \int_{c} \sin 2 z\ d z = \frac{1}{2}\ (\int_{0}^{1} \sin 2 x\ d x + i\ \int_{0}^{1} \sin 2\ (1 + i y)\ d y )\ (1)$

Kind regards

$\chi$ $\sigma$
 

nacho

Active member
Sep 10, 2013
156
thanks fellas!