(d) Demonstrate mathematical reasoning to prove that that the real-valued function х2 — у? u(х, у) %3D х3 + 2х2 - 3ху? 2у2 + (x² + y2)2 is harmonic. Construct a harmonic conjugate of u(x, y) and write down an entire function f(z) such that Re(f(z)) = u(x,y), where z = x + yi. In your working, you may use the integration formula a? – 3y? dy J (a² + y²)³ y + C (a² + y²)² for any real constant a.

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Q: 9) Find a linear function of the form f(x) = ax +b that satisfy the conditions A-4) = 8 and (3) - 9.

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Q: b) Prove that if f(z) and f(Z) are analytic in a domain I, then I is a constant function.

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