complex analysis

May 2010
2
0
hello. I would like some direction in how to approach these problems

1.Calculate the line integral
\(\displaystyle
\int\limits_{-ipi}^{ipi}({e^z-z^2}) dz
\) calculated along circle at origin & radius r=pi

2.Find the sum of the fourier series of f(x)= 1 -x on [-pi, pi]

the 1st i tried substituting x+ iy but it looks strange and the parametrization of the line, should I change to polar coordinates? Im trying to use z(t)= a.t + b

the 2nd i set f(x)= 1-sinx which makes it an odd function. but Im not sure how to solve it. like how to get values of n in sin(nt)dt.
 

chisigma

MHF Hall of Honor
Mar 2009
2,162
994
near Piacenza (Italy)
The function \(\displaystyle f(z)= e^{z} - z^{2}\) is analyitic in the whole complex plane, so that is...

\(\displaystyle \int_{\gamma} f(z)\cdot dz =0 \) (1)

... for any closed path \(\displaystyle \gamma\). That means that the integral of f(*) from \(\displaystyle -i \pi\) to \(\displaystyle + i \pi\) doesn't depend from the path connecting the limits of integration, so that we can choose the direct path along the \(\displaystyle i \omega\) axis and is...

\(\displaystyle \int_{- i \pi}^ {+ i \pi} (e^{z} - z^{2})\cdot dz = i \int_{-\pi}^{+ \pi} (e^{i \omega} + \omega^{2}) \cdot d\omega = i \int_{-\pi}^{+ \pi} (cos \omega + \omega^{2}) \cdot d\omega = \frac{2}{3} i \pi^{3}\) (2)

Kind regards

\(\displaystyle \chi\) \(\displaystyle \sigma\)
 

chisigma

MHF Hall of Honor
Mar 2009
2,162
994
near Piacenza (Italy)
Second question: is...

\(\displaystyle f(x)= \frac{a_{0}}{2} + \sum_{n=1}^{\infty} a_{n} \cos n x + b_{n} \sin n x \) (1)

... where...

\(\displaystyle a_{0} = \frac{1}{\pi}\cdot \int_{-\pi} ^{ + \pi} (1-x)\cdot dx = 1\)

\(\displaystyle a_{n} = \frac{1}{\pi}\cdot \int_{-\pi} ^{ + \pi} (1-x)\cdot \cos nx \cdot dx= 0\)

\(\displaystyle b_{n} = \frac{1}{\pi}\cdot \int_{-\pi} ^{ + \pi} (1-x)\cdot \sin nx \cdot dx= 2\cdot \frac {(-1)^{n}}{n}\) (2)

... so that is...

\(\displaystyle f(x) = 1 - 2 \sin x + \sin 2x - \frac{2}{3} \sin 3x + \dots\) (3)

Kind regards

\(\displaystyle \chi\) \(\displaystyle \sigma\)
 
May 2010
2
0
Thanks for your help. Could you give me an idea of what steps you used or did you just plug in formulas?