From: cartoaje Subject: Re: zero's of analytic function Date: Sat, 11 Mar 2000 17:04:28 -0800 Newsgroups: sci.math Summary: derivatives of periodic functions cannot have fewer zeros also sprach Jay Wiedeker: >Suppose we have an complex analytic function which is periodic of (real) >period 1, then we know that f has only a finite number of zero's in the >(real) interval [0,1). How do we know this? >The claim was made that the application of (const. + d/dz) to f cannot >diminish the number of real zero's in [0,1). How can one show this ? f has either an infinite number of zeros or no zeros on the real axis. If it has none, then your theorem follows. If it has an infinite number, suppose c=const.=0. Then you use the mean value heorem to show that f' has a zero between any two zeroes of f. Suppose c>0. Choose a zero of f (call it z) and go backwards on the real axis until you meet a zero of f' (call it m). Suppose f (m)>0 . Then f'(z)<=0. If f'(z)=0, then your theorem follows. c * f(m)+f'(m) = c*f(m)>0. c*f(z)+f'(z) = f'(z) < 0. Since f is analytic, c*f + f' has a zero between any two zeros of f. And your theorem follows. You can also prove this theorem by counting the mutiplicities, but I shall leave this as an exercise. Mihai * Sent from RemarQ http://www.remarq.com The Internet's Discussion Network * The fastest and easiest way to search and participate in Usenet - Free! ============================================================================== From: "Daniel Giaimo" Subject: Re: zero's of analytic function Date: Tue, 14 Mar 2000 23:34:54 -0800 Newsgroups: sci.math "cartoaje" wrote in message news:0121c160.3eed2646@usw-ex0101-005.remarq.com... > also sprach Jay Wiedeker: > > >Suppose we have an complex analytic function which is periodic of (real) > >period 1, then we know that f has only a finite number of zero's in the > >(real) interval [0,1). > > How do we know this? If it had an infinite number of zeroes in [0,1) then the set of zeros would have an accumulation point which implies that the function is everywhere zero. This is an easy consequence of the Uniqueness Theorem for analytic functions.