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**1. Homework Statement**

Suppose f(z) is an analytic function on domain D, and suppose that, for all z in D, we have 2*Re(f(z)) + 3*Im(f(z))=12. prove that f(z) must be a constant.

**2. Homework Equations**

**3. The Attempt at a Solution**

ok, im drawing somewhat of blank with this one but im guessing it has something to do with the partial derivatives.

since f(x +yi) = u(x,y) + i*v(x,y)

i rewrite the equations as 2*u(x,y) * 3*v(x,y) = 12

since f(z) is analytics on D, i know that

u_x' = v_y' and u_y' = - v_x'

but if I differentiate both sides of 2*(u,x) * 3*v(x,y) = 12 with respect to y and x I get a slope of 0 in each case, i.e

2*u_x' + 3*v_x' = 0

and

2*u_y' + 3*v_y' = 0

the only solution for these two equations to hold is one where f(z) is constant.

Is this correct?

any help is appreciated.

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