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Let $f:\mathbb{C}\to\mathbb{C}$ be entire function and $g:\mathbb{C}\to\mathbb{C}$ be $g(z)=f(z)-f(z+1)$. Which of the following statements are true?

a. If $f(1/n)=0$ for all positive integers $n$, then $f$ is a constant function.

b. If $f(n)=0$ for all positive integers $n$, then $f$ is a constant function.

c. If $f(1/n)= f(1/n + 1)$ for all positive integers $n$, then $g$ is a constant function.

d. If $f(n)= f(n + 1)$ for all positive integers $n$, then $g$ is a constant function.

I am stuck on this problem. Can anyone help me please?

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    @akkkk Who might "us" be?2012-12-17

3 Answers 3

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Hint 1: Use the Identity Theorem

Hint 2: $\left\{\frac{1}{n}:n\in \mathbb{N}^*\right\}$ has an accumulation point ($0$)

Hint 3: $\mathbb{C}$ is connected

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    ok.......... thanks sir....2012-12-17
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Can you think of any entire function that has zeros at regular intervals (consider periodic functions)? Simple transformations of one of those should provide a counterexample to both (b) and (d)--in particular, let $f$ be one such (appropriately transformed) function, and show that neither $f$ nor $g$ is constant.

Note: By definition of $g$, we have $f(\frac1n)=f(\frac1n+1)$ for all positive integers $n$ if and only if $g(\frac1n)=0$ for all positive integers $n.$

Now, an entire function is defined and analytic on an open connected set--namely, all of $\Bbb C$--so by the Identity Theorem, if two entire functions agree on a set $S$ that has some accumulation point in $\Bbb C$, then they are in fact the same function. Observing that any constant function (and in particular, the constant $0$ function) is entire, and that the set of reciprocals of positive integers has an accumulation point at $0$, what can you conclude about an entire function that has a zero at $\frac1n$ for every positive integer $n$? This should let you conclude that (a) is true, and (along with the Note) that (c) is true.

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Only the cases (a) and (c) are correct since the sequence has an accumulation point. The other cases do not leads to the mentioned results. Use the Identity Theorem. See this link:

http://en.wikipedia.org/wiki/Identity_theorem

In the theory of analytic functions, one have the analytic continuation principle: if two functions are equal on a set with an accumulation point, these functions are equal for all values. if $(r_{n})$ is a sequence of distinct numbers such that $ƒ(r_{n})=0$ for all $n$ and this sequence converges to a point $r$ in the domain of $D$, then $ƒ$ is identically zero on the connected component of $D$ containing $r$.

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    @pankaj: you don't need to be "sorry". just read up on the identity theorem, and ask us about that instead of trying to answer these questions.2012-12-17