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+--- Thread: exp^[1/2](x) uniqueness from 2sinh ? (/showthread.php?tid=879)
exp^[1/2](x) uniqueness from 2sinh ? - tommy1729 - 06/03/2014
A possible uniqueness critertion for exp^[1/2](x) ?
For x > 1 and any integer n >= 0 :
1) e/n! > d^n exp^[1/2](x)/d^n x @ x = 1 > 0.
2) 2sinh^[1/2](x) + d 2sinh^[1/2](x)/dx - exp(-x) > exp^[1/2](x) > 2sinh^[1/2](x).
( 2sinh^[1/2](x) is computed with the koenigs function )
3) exp^[1/2](z) is holomorphic for Re(z) > 1/2.
If the uniqueness fails the question is if the conditions are too strong or too weak.
And if it can be improved.
regards
tommy1729
RE: exp^[1/2](x) uniqueness from 2sinh ? - tommy1729 - 06/03/2014
Hmm The conditions must fail because they imply that exp^[1/2](x) is entire which it is not.
Not sure how to bound the derivatives then ...
reduce condition 1) to d^n exp^[1/2](x)/d^n x @ x = 1 > 0 ?
regards
tommy1729