CSIR NET Mathematics 2022| Complex Analysis (10 NOV)

Let f(z)=cos.then for f(z),z= 0 is

Let u(x, y) = x³ + ax²y + bx y² + 2y³ be a harmonic function v(x, y) its harmonic conjugate. If u(0, 0) = 1 then |a –b + v(1, 0)| equal to

Let C be the circular contour taken counterclockwise. The line integer  is equal to

Given a real number a > 0, consider the triangle  with vertices 0, a, a + ia. If  is given the counter clockwise orientation, then the contour integral  (with Re(z) denoting the real part of z) is equal to

Which of the following function has removable singularity at z =0