Question

Check whether the function f(z)=(2z-z^2) is harmoic or not?

Answer 1

See below, this is harmonic

Explanation:

`f(z) = 2(x + i y ) - (x + i y)^2`

`= underbrace( 2x- x^2 + y^2)_(u(x,y)) + i \ underbrace (2 y( 1 - x))\_(v(x,y))`

The function will be harmonic if both its real and its imaginary component satisfy the Laplace Equation: ie `\Delta u =0`
and `\Delta v =0`

`u_("x") = 2 - 2x qquad u_("y") = 2y`

`u_("xx") = - 2 qquad u_("yy") = 2`

• `implies u_("xx") + u_("yy") = 0`

`v_("x") = - 2y qquad v_("y") = 2(1 - x)`

`v_("xx") = 0 qquad v_("yy") = 0`

• `implies v_("xx") + v_("yy") = 0`