For example, what is the antiderivative of $t^{0.3} \cos (t)$? Of course this is the real part of $t^{0.3} e^{it}$
Comment: Originally I was thinking about an antiderivative. But a definite integral also helps.
Asked
Active
Viewed 83 times
1
-
2What is the range of integration? Anyway, look up "Mellin transform", it should be interesting for you. – Giuseppe Negro Mar 30 '23 at 11:08
-
Thank you! I've edited the question – teepung Mar 30 '23 at 11:15
-
Consider $\Re \int t^a e^{it},dt$ – Claude Leibovici Mar 30 '23 at 11:53
-
@ClaudeLeibovici, i know it's the real part of that, but there's the same question about your integral... I think negro's suggestion about Mellin transform is a good one. – teepung Mar 30 '23 at 14:46
-
Yes for sure. You also have thé incomplète gamma function – Claude Leibovici Mar 30 '23 at 14:54
-
1There is a solution if you accept the incomplete gamma function. You can find a step-by-step guide for this, e.g. here. – Kevin Dietrich Mar 30 '23 at 15:03
-
1As mentioned above regarding the Mellin transform, this has been adressed here: https://math.stackexchange.com/questions/1912258/provide-us-a-detailed-explanation-to-calculate-the-mellin-transform-of-a-cosine – AxelT Mar 30 '23 at 17:45
-
1This is the Bohmer integral – Тyma Gaidash Apr 06 '23 at 15:53
1 Answers
1
Incomplete Gamma Function $$ \begin{aligned} & \int t^\alpha \cos t d t \\ = & \operatorname{Re} \int_0^t x^\alpha e^{-i x} d x \\ \stackrel{ i x \rightarrow x }{=}& \operatorname{Re} \int_0^{t i}\left(\frac{x}{i}\right)^\alpha e^{-x} \frac{d x}{i} \\ = & \operatorname{Re}\left[(-i)^{\alpha+1} \int_0^{it} x^\alpha e^{-x} d x\right] \\ = & \operatorname{Re}\left[(-i)^{\alpha+1}(\Gamma(\alpha+1)-\Gamma(\alpha+1, i t))\right] \end{aligned} $$ Wish it helps!
Lai
- 20,421
-
Hi Lai, I'm confused about your last step. Looking at this wiki page about incomplete gamma function: https://en.wikipedia.org/wiki/Incomplete_gamma_function The integral is defined over the real numbers (represented by t on the wiki page, and the upper bound x should be a real number), what do you mean by 'i t' as an upper bound? I'm not familiar with the gamma function, just asking. – teepung Apr 05 '23 at 16:22