List of top Engineering Mathematics Questions on Fourier series asked in TS PGECET

If \( u = f(r) \), \( r^2 = x^2 + y^2 + z^2 \), then \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = ? \]
If for a progression, sum of the first $n$ terms is $2n^2+5n+6$ and $T_n$ is the $n^{th}$ term of the progression, then $\sum_{k=1}^{10} T_{3k} = $?

Let $f(x) = \begin{cases} h(x), & 0<x<C \\-h(-x), & -C<x<0 \end{cases}$ and $f(x+2C)=f(x) \; \forall x \in \mathbb{R}$. If the Fourier series of $f(x) = \sum_{n=0}^\infty \left(a_n \cos\frac{n\pi x}{C} + b_n \sin\frac{n\pi x}{C}\right)$ then $\sum_{n=0}^\infty a_n b_n =$

Let \[ f(x) = \begin{cases} 1 + \dfrac{2x}{\pi}, & -\pi \le x \le 0
1 - \dfrac{2x}{\pi}, & 0<x \le \pi \end{cases} \] The constant term of the Fourier series of \(f(x)\) is:
If $f(x) = \begin{cases} -\pi, & \text{in } -\pi<x<0 \\ x, & \text{in } 0<x<\pi \end{cases}$ then the constant term of the Fourier series is
If \[ f(x) = \begin{cases} -\pi, & -\pi<x<0 \\ x, & 0<x<\pi \end{cases} \] then the constant term of the Fourier series is