Here is an example of misunderstandings about analytic functions. Explain the pr

Question

Here is an example of misunderstandings about analytic functions. Explain the problem(s):

E. Picard's Great Theorem pertains to the behavior of a complex function in the neighborhood of an essential singuarity. Picard's Lesser Theorem states that "A transcendental entire function actually takes on every complex value with at most one exception outside every circle." Something must be wrong with the following argument? What is it?

NOTE: Picard's Theorem is false, since the function f(z)=e^e^z has two exceptions. This function is certainly an entire transcendental function. If we consider the circle |z-3|=1 then we know e^z is never zero, so that e^e^z is never 1, since e^e^z is also an exponential function it is never equal to 0. Therefore there are actually two values (namely 0- and 1) that are not taken on.

Explanation / Answer

The error in the argument is that value of f(z) must be considered I.e., at Z= -10000000

F(z) =1

The value of f(z) lies in the circle only... The function is transcendental and entire function so at z=0 the function has one and only one exception

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