Verifying Entireness of Analytic Functions Using Cauchy-Riemann Theory

[Benzoate]

Homework Statement

Apply rules of Cauchy-riemann theory to verify that each of these functions is entire:

f(z)=3*x+y+i(3y-x)

Homework Equations

u_x=v_y, u_y=-v_x

The Attempt at a Solution

u(x,y)=3x+y
v(x,y)=3y-x

u_x=3
v_y=3
u_y=1
-v_y=1
I know that a function is analytic at each point, then the function is entire. How would I show that the function is analytic at each pt.?

 

[Dick]

You've shown it satisfies CR everywhere. The only thing that could prevent it from being entire is if it has a removable singularity someplace. Does it? Does it have ANY singularities?