Question Solved1 Answer Answer Problem 2. Make sure its clear to read, Show all work. Problem 2. State why a composition of two entire functions is entire. Also, state why any linear combination \( c_{1} f_{1}(z)+c_{2} f_{2}(z) \) of two entire functions is entire, for \( c_{1}, c_{2} \in \mathbb{C} \).
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Answer Problem 2. Make sure its clear to read, Show all work.
Transcribed Image Text: Problem 2. State why a composition of two entire functions is entire. Also, state why any linear combination \( c_{1} f_{1}(z)+c_{2} f_{2}(z) \) of two entire functions is entire, for \( c_{1}, c_{2} \in \mathbb{C} \).
More Transcribed Image Text: Problem 2. State why a composition of two entire functions is entire. Also, state why any linear combination \( c_{1} f_{1}(z)+c_{2} f_{2}(z) \) of two entire functions is entire, for \( c_{1}, c_{2} \in \mathbb{C} \).
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1st part: Ltf_(1) and f_(2) be two entire functions. since both f_(1) and f_(2) are entire function, both are well-defined on the complen plame (C and also hoth heve derivatives for all z inCsince f_(1) and f_(2) are well defined the function f_(1)@f_(2)(z) is well defined by using chain rule of differentiation for evory z inC (f_(1)@f_(2))^(')(z)=(d)/(dz)(f_(1)(f_(2)(z)))=f_(1)^(')(f_(2)(z))(f_(2)^(')(z)) and ... See the full answer