Engineering Mat...

The amplitude of \({e^{{e^{ - i\theta }}}}\) is equal to

The value of ‘b’ for which u(x, y) = e^bx cos 5y is harmonic

If the function defined by f(z) = (x² + axy + by²) + i(cx² + dxy + y²) is analytic, then (a + b + c + d) = ______

The Taylor series expansion of 3 sin x + 2 cos x is

Let f(z) = (x² + y²) + i2xy and g(z) = 2xy + i(y² – x²) for z = x + iy ϵ C. Then, in the complex plane C.

A complex function f(z) = u (x, y) + i v (x, y) and its complex conjugate f*(z) = u(x, y) – i v(x, y) are both analytic in the entire complex plane, where z = x + i y and \({\rm{i}} = \sqrt { - 1}\). The function f is then given by

The value of the integral \(\mathop \smallint \nolimits_c \frac{{dx}}{{{z^2} + 9}}\) where c is the circle |z - 3i| = 4 is

In the Laurent expansion of \(f\left( z \right) = \frac{1}{{\left( {z - 1} \right)\left( {z - 2} \right)}}\) valid in the region 1 < |z| < 2, the co-efficient of \(\frac{1}{{{z^2}}}\) is

If u(x, y) = e^x (x cos y – y sin y) then f(z) is

The conjugate harmonic function v(x, y) of u(x, y) = In (x² + y²) is