A periodic function f(x) = x for -π < x < +π has the Fourier series representation \[f\left( x \right) = \sum\limits_{n = 1}^\infty {\left( { - \frac{2}{n}} \right){{\left( { - 1} \right)}^n}\sin nx.} \]
Using this, one finds the sum \[\sum\limits_{n = 1}^\infty {{n^{ - 2}}} \]   to be

The contour integral $$\oint {\frac{{dz}}{{{z^2} + {a^2}}}} $$   is to be evaluated on a circle of radius 2a centred at the origin. It will have contributions only from the points

Which of the following functions of the complex variable z is not analytic everywhere?

Consider a vector \[\overrightarrow {\bf{p}} = 2{\bf{\hat i}} + 3{\bf{\hat j}} + 2{\bf{\hat k}}\]     in the coordinate system \[\left( {{\bf{\hat i}},\,{\bf{\hat j}},\,{\bf{\hat k}}} \right).\]   The axes are rotated anti-clockwise about the Y-axis by an angle of 60°. The vector \[\overrightarrow p \] in the rotate coordinate system \[\left( {{\bf{\hat i}},\,{\bf{\hat j}},\,{\bf{\hat k}}} \right)\]   is

The value of the integral $$\oint\limits_C {\frac{{{e^z}\sin z}}{{{z^2}}}} dz,$$   where the contour C is the unit circle: |z - 2| = 1, is