g1990 said:lets see, if f(z) never equals w, then g(z) is analytic everywhere on C. Then, as z approaches infinity, g(z) approaches 0 because f(z) approaches infinity. I'm not sure why this is a contradiction.
g1990 said:Lets see... the modulus of f is definitely bounded from below by 0. If I could get it to be bounded from above that would be a contradiction b/c of the maximum modulus principle. I'm not sure how to do that though. You could say that if z if large, f is above a certain value so that g is below a certain value. Then, f has a minimum value on the remaining compact set, so that g has a maximum on that compact set. Then the max of the two would be the bound for g. Is that right?
g1990 said:Oh, maybe I need that N>|w|?
g1990 said:Okay, so if I let N>2|w|, then my argument will work?
To prove that a function f(z) is entire, you need to show that it is analytic at every point in the complex plane. This means that the function must have a derivative at every point and that this derivative must also be continuous. Additionally, you need to show that the function is defined and continuous at every point.
An entire function is a complex function that is defined and analytic at every point in the complex plane. This means that the function has a derivative at every point and that this derivative is also continuous. In other words, an entire function is a complex function that is well-behaved and defined at every point in the complex plane.
To prove that the image of an entire function f(z) is all of C (the complex plane), you need to show that for every point z in C, there is a corresponding point w in the image of f(z). This means that the function is surjective, or onto. Additionally, you need to show that the function is defined and continuous at every point in the complex plane.
There are several techniques that can be used to prove that a function is entire. Some common techniques include using the Cauchy-Riemann equations, the Cauchy integral formula, the power series expansion, and the identity theorem. It is important to carefully choose which technique to use based on the specific function and problem at hand.
No, a function cannot be entire but not analytic. The definition of an entire function includes the requirement that the function is analytic at every point in the complex plane. Therefore, if a function is entire, it must also be analytic at every point.