solve the following problems

B5. Let fn(2) be a sequence of functions holomorphic in the connected open set ? and assume they converge uniformly on every compact subset of 1. Show that the sequence of derivatives fh(2) also converges uniformly on every compact subset of !. B6. Find a conformal map from the unbounded region outside the disks {|=+1| 0 the function (Laplace transform) g ( 2 ) := f(the -= at is holomorphic for Re(=} > 0.Traditional Problems [10 points each (70 points total)] B1. Assume f(2) is meromorphic for all (=| 0.Short Answer Problems [5 points each (20 points total)] Al. If f(=) is an entire function with If(=)| 2 1 everywhere, what can you conclude about f? Justify your assertions. A2. If f(=) is an entire function and f(x + 2x) = f(x) for all real x, does f(= + 2x) = f(2) for all complex =? Proof or counterexample. A3. The function = 22+32 -4 has a power series expansion in a neighborhood of the origin. What is its radius of convergence? Justify your assertion. A4. Assume the entire function f(2) has no zeroes on any of the circles |:| = n, n = 1,2,3,... and also that fum They de + from They de. " = 1,2.3. Is this function transcendental? Proof or counterexample