User petern - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:13:22Z http://mathoverflow.net/feeds/user/30924 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119857/the-real-and-imaginary-parts-of-the-incomplete-gamma-function-for-second-argument The real and imaginary parts of the Incomplete Gamma function for second argument being purely imaginary Petern 2013-01-25T16:59:10Z 2013-05-05T01:22:00Z <p>Dear all, I am looking for explicit (at least more explicit than the original expression) for</p> <p>1) Re$(\Gamma(a, i\omega))$ </p> <p>as well as</p> <p>2) Im$(\Gamma(a, i\omega)),$</p> <p>where i Re and Im denote the real and imaginary part, and $\Gamma(a, i\omega)$ is the Incomplete Gamma function with the arguments $a$ and $i\omega$. The letter $i$ denotes imaginary unit, $a>0$ is a real number and $\omega$ is also a real number.</p> <p>I would presume that 1) and 2) could be written as functions of $\Gamma(a, \omega)$, however by application of change of variables in the integral definition of the Incomplete Gamma function did not succeed for me.</p> <p>Thank you in advance for any kind of input.</p> http://mathoverflow.net/questions/111565/eigenfunction-of-local-fractional-derivative/119958#119958 Answer by Petern for Eigenfunction of local fractional derivative Petern 2013-01-26T19:58:09Z 2013-01-26T20:07:41Z <p>From my point of view, your definition is strange, or at least it has a contra-intuitive feature: because $x$ and $\delta$ should have the same unit (e.g. some length unit), $\delta^\alpha$ will have the unit e.g. length$^\alpha$. Then $\tilde D^a f(a)$ will have the unit "meter$^{1-\alpha}$". But maybe this is OK? Please correct me if I'm wrong.</p> <p>(This was meant as a comment and not an answer. Now that I posted this as an answer, I however after deleting it cannot post a comment... Therefore I un-delete the answer, but reader please consider it as a comment :-) )</p> http://mathoverflow.net/questions/111999/local-fractional-derivative-that-doesnt-vanish-on-differentiable-functions/119859#119859 Answer by Petern for Local fractional derivative that doesn't vanish on differentiable functions Petern 2013-01-25T17:04:18Z 2013-01-25T17:04:18Z <p>I'm not sure, but maybe you could investigate the Yang local fractional derivative?</p> http://mathoverflow.net/questions/103112/are-there-analogous-theorems-and-or-techniques-for-solving-fractional-differentia/119858#119858 Answer by Petern for Are there analogous theorems and/or techniques for solving fractional differential equations involving the Riesz Derivative? Petern 2013-01-25T17:01:58Z 2013-01-25T17:01:58Z <p>The fractional Riez derivative can be written as a sum of Caputo or Reimann-Liouville fractional derivatives, so maybe this could help you?</p> http://mathoverflow.net/questions/119857/the-real-and-imaginary-parts-of-the-incomplete-gamma-function-for-second-argument/119890#119890 Comment by Petern Petern 2013-02-07T22:18:48Z 2013-02-07T22:18:48Z Robert, your hint pointed me towards what I was looking for: the &quot;generalized sine and cosine functions.&quot; More infor can be retreived e.g. from here: <a href="http://dlmf.nist.gov/8.21" rel="nofollow">dlmf.nist.gov/8.21</a> http://mathoverflow.net/questions/119857/the-real-and-imaginary-parts-of-the-incomplete-gamma-function-for-second-argument/119890#119890 Comment by Petern Petern 2013-01-26T10:55:31Z 2013-01-26T10:55:31Z Thanks a lot Robert for the hint. I will explore this further after the week-end. I even suspect that the Lommel S1 function might possibly be written in terms of the Incomplete Gamma function, but this I need to look up in detail. Again thank you.