Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

suppose that $f$ is an entire function such that $f'(0)=0$ and $f''(1+1/n)=7-3/n$ for all $n$ natural number. we have to find all $f$ that satisfies these properties.

What I have done is: define $g(z)=f''(z)-10+3z$, so $g(1+1/n)=0$ so uniqueness theorem implies that $f''(z)=10-3z$ and therefore $f'(z)=10z-3z^2/2 +a$, As $f'(0)=0$ so $f(z)=5z^2-z^3/2+b(constant)$ is my solution is correct?

share|improve this question
    
What is the uniqueness theorem? –  William Jun 10 '12 at 5:47
    
if zero set has limit point in the domain of the analytic function then it is identically zero there. –  Une Femme Douce Jun 10 '12 at 5:50
    
I think you meant $f''(z)=10-3z$. –  Gerry Myerson Jun 10 '12 at 6:38
    
ah yes u r right... –  Une Femme Douce Jun 10 '12 at 6:41
    
OK, I have edited in the correction for you. –  Gerry Myerson Jun 10 '12 at 12:34

1 Answer 1

up vote 2 down vote accepted

The solution is correct: since the equality $f''(z)=10-3z$ holds on a set with a limit point, it holds for all $z\in\mathbb C$. By integration, using the condition $f'(0)=0$ it follows that $f(z)=5z^2-z^3/2+\text{const}$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.