# complex analysis

#### journyman

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
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
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

$$\displaystyle \chi$$ $$\displaystyle \sigma$$

#### journyman

Thanks for your help. Could you give me an idea of what steps you used or did you just plug in formulas?