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Generalization of Carlson theorem for double zeroes.
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Carlson's theorem states that a function $f(z)$ has a zero at all nonnegative integer values, and has exponential growth in the real and imaginary axes, then it is identically zero.
I was looking for a possible generalization of it which requires double zero at all integer values. Is it possible to relax any of the other assumptions?
Note: $sin(pi z)$ function serves as a counterexample for the original theorem, if the exponential growth was not bounded (strictly) by $pi$. However, given that we now have double zeroes, can this bound be relaxed and a new one found?
Also, is there any example which is better than $sin^2 (pi z)$ for this theorem?
complex-analysis maximum-principle
asked Jul 28 at 4:17
Aaditya Salgarkar
1
add a comment |Â
up vote
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Carlson's theorem states that a function $f(z)$ has a zero at all nonnegative integer values, and has exponential growth in the real and imaginary axes, then it is identically zero.
I was looking for a possible generalization of it which requires double zero at all integer values. Is it possible to relax any of the other assumptions?
Note: $sin(pi z)$ function serves as a counterexample for the original theorem, if the exponential growth was not bounded (strictly) by $pi$. However, given that we now have double zeroes, can this bound be relaxed and a new one found?
Also, is there any example which is better than $sin^2 (pi z)$ for this theorem?
complex-analysis maximum-principle
asked Jul 28 at 4:17
Aaditya Salgarkar
1
You might start with a complete and correct statement of the theorem...
– David C. Ullrich
Jul 28 at 15:09
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Carlson's theorem states that a function $f(z)$ has a zero at all nonnegative integer values, and has exponential growth in the real and imaginary axes, then it is identically zero.
I was looking for a possible generalization of it which requires double zero at all integer values. Is it possible to relax any of the other assumptions?
Note: $sin(pi z)$ function serves as a counterexample for the original theorem, if the exponential growth was not bounded (strictly) by $pi$. However, given that we now have double zeroes, can this bound be relaxed and a new one found?
Also, is there any example which is better than $sin^2 (pi z)$ for this theorem?
complex-analysis maximum-principle
asked Jul 28 at 4:17
Aaditya Salgarkar
1
Carlson's theorem states that a function $f(z)$ has a zero at all nonnegative integer values, and has exponential growth in the real and imaginary axes, then it is identically zero.
I was looking for a possible generalization of it which requires double zero at all integer values. Is it possible to relax any of the other assumptions?
Note: $sin(pi z)$ function serves as a counterexample for the original theorem, if the exponential growth was not bounded (strictly) by $pi$. However, given that we now have double zeroes, can this bound be relaxed and a new one found?
Also, is there any example which is better than $sin^2 (pi z)$ for this theorem?
complex-analysis maximum-principle
asked Jul 28 at 4:17
Aaditya Salgarkar
1
asked Jul 28 at 4:17
Aaditya Salgarkar
1
asked Jul 28 at 4:17
Aaditya Salgarkar
1
asked Jul 28 at 4:17
Aaditya Salgarkar
1
1
You might start with a complete and correct statement of the theorem...
– David C. Ullrich
Jul 28 at 15:09
add a comment |Â
You might start with a complete and correct statement of the theorem...
– David C. Ullrich
Jul 28 at 15:09
You might start with a complete and correct statement of the theorem...
– David C. Ullrich
Jul 28 at 15:09
You might start with a complete and correct statement of the theorem...
– David C. Ullrich
Jul 28 at 15:09
add a comment |Â
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You might start with a complete and correct statement of the theorem...
– David C. Ullrich
Jul 28 at 15:09