Fourier Series Value Calculation

We need to compute $F(-\frac{\pi}{2}) + F(0)$, where $F(x)$ is the Fourier series of the given piecewise function $f(x)$.

The function is defined as:

with period $2\pi$. The Fourier series is $F(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))$.

Fourier Coefficients Calculation

$a_0$ Coefficient

• $a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) dx = \frac{1}{\pi} \int_{-\pi}^{0} (\pi + x) dx$
• $a_0 = \frac{1}{\pi} \left[ \pi x + \frac{x^2}{2} \right]_{-\pi}^{0} = \frac{1}{\pi} [0 - (-\pi^2 + \frac{\pi^2}{2})] = \frac{\pi}{2}$

$a_n$ Coefficients

• $a_n = \frac{1}{\pi} \int_{-\pi}^{0} (\pi + x) \cos(nx) dx$
• Integration by parts yields: $a_n = \frac{1}{\pi} \left[ \frac{1 - (-1)^n}{n^2} \right]$.
• $a_n = 0$ for even $n$, and $a_n = \frac{2}{\pi n^2}$ for odd $n$.

$b_n$ Coefficients

• $b_n = \frac{1}{\pi} \int_{-\pi}^{0} (\pi + x) \sin(nx) dx$
• Integration by parts yields: $b_n = \frac{1}{\pi} \left[ -\frac{\pi}{n} \right] = -\frac{1}{n}$.

Fourier Series Evaluation

The Fourier series is $F(x) = \frac{\pi}{4} + \sum_{k=0}^{\infty} \frac{2}{\pi(2k+1)^2} \cos((2k+1)x) - \sum_{n=1}^{\infty} \frac{1}{n} \sin(nx)$.

Value at $F(-\frac{\pi}{2})$

• At $x = -\frac{\pi}{2}$, the cosine terms are zero since $\cos((2k+1)(-\frac{\pi}{2})) = 0$.
• $F(-\frac{\pi}{2}) = \frac{\pi}{4} - \sum_{n=1}^{\infty} \frac{1}{n} \sin(-\frac{n\pi}{2}) = \frac{\pi}{4} + \sum_{n=1}^{\infty} \frac{1}{n} \sin(\frac{n\pi}{2})$
• The sum $\sum_{n=1}^{\infty} \frac{1}{n} \sin(\frac{n\pi}{2}) = 1 - \frac{1}{3} + \frac{1}{5} - ... = \frac{\pi}{4}$ (Leibniz formula).
• Therefore, $F(-\frac{\pi}{2}) = \frac{\pi}{4} + \frac{\pi}{4} = \frac{\pi}{2}$.

Value at $F(0)$

• The function $f(x)$ has a jump discontinuity at $x=0$.
• $F(0)$ converges to the average of the left and right limits: $\frac{f(0^-) + f(0^+)}{2}$.
• $f(0^-) = \lim_{x \to 0^-} (\pi + x) = \pi$.
• $f(0^+) = \lim_{x \to 0^+} 0 = 0$.
• $F(0) = \frac{\pi + 0}{2} = \frac{\pi}{2}$.

Final Sum Calculation

Summing the values:

• $F(-\frac{\pi}{2}) + F(0) = \frac{\pi}{2} + \frac{\pi}{2} = \pi$.