= Let f(x) e be the exponential function defined on the real numbers x € R. Using the theorems we learned from this course to prove that F₁(z) = e² = eª (cos(y) + i sin(y)) Vz= x+iy € C is an entire function. That is, F(z) is analytic on the entire finite complex plane C. Check that e* cos(y) and eª sin(y) are harmonic functions by direct calculation. Notice that F₁ (2) is an extension of f(x). Find all the possible extensions F(2): C → C of f(x): R → R that are also entire functions. Show your calculation and argument.

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