Q4 TUT 0101

[Victor Ivrii]

Evaluate the definite trigonometric integral making use of the
technique of Examples 6 and 7 in Section 2.3 of the Textbook:
$$
\int_{0}^{2\pi} \frac{d\theta}{a+b\cos(\theta)}, \qquad a>b.
$$

[Jeffery Mcbride]

\begin{equation*}
cos\ \theta \ =\ \frac{1}{2} \ \left( z\ +\ \frac{1}{z} \ \right)\\
\\
d\theta \ =\ \frac{dz}{iz}\\
\\
\frac{d\theta }{a\ +\ bcos\theta } \ =\ \frac{dz}{iz\ \left( a\ +\ \frac{bz}{2} \ +\ \frac{b}{2z} \ \right)}\\
\\
=\ \frac{dz}{\frac{i}{2}\left( bz^{2} \ +\ 2az\ +\ b\right)}\\
\\
=\ \frac{2\ dz}{\ i\left( bz^{2} \ +\ 2az \ +\ b\right)}\\
\\
Complete\ the\ square:\\
=\ \frac{-2a \ \pm \sqrt{4a^{2} \ -\ 4b^{2}}}{2b} \ \\
\\
=\ \frac{-a \ \pm \sqrt{a^{2} \ -\ b^{2}}}{b}\\
\\
\left( z\ +\frac{a}{b} \ +\sqrt{\frac{a^{2}}{b^{2}} \ -\ 1} \ \right)\left( z\ +\frac{a}{b} \ -\sqrt{\frac{a^{2}}{b^{2}} \ -\ 1} \ \right)\\
\\
\ ( z\ -\ i\left( -\ \frac{a}{b} \ -\sqrt{\frac{a^{2}}{b^{2}} \ -\ 1} \ \right) \ is\ outside\ of\ |z|\ =\ 1\ =\ z\ -\ q\\
\\
( z\ +\ i\left( \ \frac{a}{b} \ \ -\sqrt{\frac{a^{2}}{b^{2}} \ -\ 1} \ \right) \ is\ inside\ of\ |z|\ =\ 1\ =\ z\ -\ p\\
\\
So\ Cauchy's\ formula\ gives\ us\\
\frac{1}{2\pi i} \ \int _{|z|\ =\ 1} \ \frac{dz}{( z\ -\ q)( z\ -\ p)} \ \ =\ \frac{1}{( p\ -\ q)} \ =\ \frac{1}{2ib\ \sqrt{\frac{a^{2}}{b^{2}} \ -\ 1}}\\
=\frac{1}{2i\ \sqrt{a^{2} \ -\ b^{2}}}\\
\\
\\
But\ the\ integral\ is\ just\\
\\
\frac{1}{2\pi i}\int _{|z|\ =\ 1} \ \frac{dz}{( z\ -\ q)( z\ -\ p)}\\
\\
=\ \frac{1}{2\pi i}\int ^{2\pi }_{0} \ \frac{ie^{i\theta } d\theta }{2ie^{i\theta }( a\ +\ bcos\theta )}\\
\\
=\ \frac{1}{4\pi i}\int ^{2\pi }_{0} \ \frac{d\theta }{( a\ +\ bcos\theta )}\\
\\
so,\\
\\
\int ^{2\pi }_{0} \ \frac{d\theta }{( a\ +\ bcos\theta )} \ =\ \frac{4\pi i}{2i\sqrt{a^{2} -b^{2}}}\\
\\
=\frac{2\pi }{\sqrt{a^{2} \ -\ b^{2}}}
\end{equation*}

[Victor Ivrii]

If you want to put text inside of math formula, do it through command \text{  } :

Code: [Select]

z=1+i \text{ is inside of } \{|z|\colon |z|<4\}producing
$$z=1+i \text{ is inside of } \{|z|\colon |z|<4\}$$

Note: \text is treated differently by LaTeX and MathJax because LaTeX compiles the whole document, in particular, what is inside of \text{  } while MathJax just leaves it as is (except, html commands inside of math sabotage MathJax).

MathJax does not provide a command for text "between the lines" while in LaTeX in some environments \intertext{   } works