5.* There exists an entire function F with the following "universal property: given any entire function h, there is an i

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5 There Exists An Entire Function F With The Following Universal Property Given Any Entire Function H There Is An I 1 (11.15 KiB) Viewed 31 times

5.* There exists an entire function F with the following "universal property: given any entire function h, there is an increasing sequence {Nx}=1 of positive integers, so that lim F(z + Nx) = n(z) 100 uniformly on every compact subset of C.

(a) Let P1, P2, ... denote an enumeration of the collection of polynomials whose coefficients have rational real and imaginary parts. Show that it suffices to find an entire function F and an increasing sequence {M} of positive integers, such that (17) |F(z) - pulz - M.)<- whenever z € Dry where Dr denotes the disc centered at M. and of radius n. (Hint: Given h entire, there exists a sequence {ns} such that lim - Pre(z) = n(z) uni- formly on every compact subset of C.] (b) Construct F satisfying (17) as an infinite series 20 F(x) = n(2) n=1 - where un(x) = príz - Mneon(2-Ma)", and the quantities en > 0 and Mn> O are chosen appropriately with an 0 and Mn. [Hint: The function vanishes rapidly as Izl > in the sectors {| arg z| </4 - 8) and { * - arg 2| <*/4-8}.] In the same spirit, there exists an alternate "universal" entire function with the following property: given any entire function h, there is an increasing sequence {Nx} of positive integers, so that lim DNG(z) = A(z) k+00 uniformly on every compact subset of C. Here D'G denotes the ith complex) derivative of G.