$u=\dfrac{x}{2}log(x^2-y^2)-y\ tan^{-1}\Bigg(\dfrac{y}{x}\Bigg)+sin\ x\ cos\ hy$ Let, $f(x)=u+iv$ be an analytic function with real part.  We have,  $\phi_1(x, y)=\dfrac{\partial u}{\partial x}=\dfrac{x}{2}.\dfrac{2x}{x^2+y^2}+\dfrac{1}{2}\ log(x^2+y^2)-y\dfrac{1}{1+\Bigg(\dfrac{y}{x}\Bigg)^2}\Bigg(\dfrac{-y}{x}\Bigg)+cosx. cos\ hy$ $\phi_2(x, \ y)=\dfrac{\partial u}{\partial y}=\dfrac{xy}{x^2+y^2}+\dfrac{y}{x^2+y^2}-tan^{-1}\Bigg(\dfrac{y}{x}\Bigg)+sin\ x.\ sin\ hy$ Put x = z and y = 0, we get,  $\phi_1(z, 0)=1+\dfrac{1}{2}log\ z^2+cos\ z$ $\phi_2\ (z, 0)=0$ $\phi_1\ (z, 0)=1+log\ z^2+ cos\ z$ **By Milne Thompson Method, the analytic function f(z) is given as;** $f(z)=\int \phi_1(z, \ 0)dz-i\int \phi_2(z, \ 0)dz+c=\int (1+log\ z^2+cos\ z)dz+c$ $f(z)=z+[z\ logz -z]+sin\ z +c$ $f(z)=z+z\ logz-z\ sinz+c$ $f(z)=zlogz+sinz +c\ is\ the\ required\ analytic\ function. $