Timeline for Question on the integral $\int_{-\infty}^{\infty} e^{a x^4+b x^3+c x^2+d x+f}\,dx$

8 events

when toggle format	what		by	license	comment
Feb 20, 2020 at 22:39	history	edited	Abdelmalek Abdesselam	CC BY-SA 4.0	added 59 characters in body
Feb 20, 2020 at 14:51	history	edited	Abdelmalek Abdesselam	CC BY-SA 4.0	added 534 characters in body
Aug 9, 2017 at 9:22	comment	added	sam		I have post my question on mathoverflow.net/questions/278334/…. Thanks for your help.
Aug 1, 2017 at 13:16	comment	added	Abdelmalek Abdesselam		@sam: Is this a research problem or a homework question? It might be worth formulating it precisely with all the hypotheses etc. as a separate MO question.
Aug 1, 2017 at 6:59	comment	added	sam		Actually I want to solve this problem: suppose $f(x)$ satisfies $\int_{-\infty}^\infty f(x)dx=1,\int_{-\infty}^\infty xf(x) dx=0 ,\int_{-\infty}^\infty x^2 f(x) dx = \theta$, find $f(x)$ such that the integral $\int f(x) \ln f(x) - f(x)dx $ reaches its maximum. It is easy to see $f(x) =e^{\alpha + \beta x + \gamma x^2}$ and we can find the values for $\alpha,\beta,\gamma$ using the Guassian integral. However if we add the constraint $\int_{-\infty}^\infty x^3 f(x) dx = q$, then $f(x) =e^{\alpha + \beta x + \gamma x^2 + \zeta x^3} $. But how to find the explicit values for the coefficients?
Jul 30, 2017 at 18:03	history	edited	Abdelmalek Abdesselam	CC BY-SA 3.0	added 1 character in body
Jul 30, 2017 at 17:51	history	edited	Abdelmalek Abdesselam	CC BY-SA 3.0	added 503 characters in body
Jul 30, 2017 at 17:44	history	answered	Abdelmalek Abdesselam	CC BY-SA 3.0