Complex Analysis Entire Functions

Thread starter bballife1508
Start date Dec 14, 2010
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Analysis Complex Complex analysis Functions

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An entire function f(z) that satisfies |f(z)| ≤ R for |z| = R implies that all derivatives at zero, f''(0), f'''(0), and higher, are equal to zero. Consequently, this leads to the conclusion that f(0) must also be zero. Two examples of such functions include the zero function and the function f(z) = e^(-z^2), both of which meet the given conditions. The discussion emphasizes the implications of the growth restrictions on entire functions and their derivatives. Overall, the properties of entire functions are crucial in understanding their behavior at the origin.

Dec 14, 2010

bballife1508

Let f(z) be an entire function such that |f(z)| less that or equal to R whenever R>0 and |z|=R.

(a)Show that f''(0)=0=f'''(0)=f''''(0)=...

(b)Show that f(0)=0.

(c) Give two example of such a function f.

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Dec 14, 2010

snipez90

Hint: look back through your old post history. I already explained one tool you could use to approach these problems.

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