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 Feb1 comment How to compute $\int_{0}^{\infty}dx\:\frac{\exp(-ax^2+bx)}{x+1}\:\text{ for }\: a>0, b\in \mathbb{C}$? Thanks. But I don't want an answer as a power series in b. Actually I am trying to see the behaviour of $\displaystyle\sum_{k=1}^{\infty}\displaystyle\int_{0}^{\infty}dx\: \frac{\exp(-a(x+1)^2+2\pi i k x)}{x+1}$ for small values of $a$. Expanding in $b=-2a+2\pi i k$ doesn't seem helpful in this problem. Dec8 comment Proof of a theorem about oscillation You are right. I see it now. Thanks. Dec8 comment Proof of a theorem about oscillation Can you include this calculation in your answer ? I am not sure if the method of variation of parameters applies to nonlinear equations Dec8 comment Proof of a theorem about oscillation Hmm.. but I can't see how the derivative of second equation is same as the first? Dec8 comment Proof of a theorem about oscillation I think the difference of x and y satisfies $\displaystyle \frac {d (x-y)}{dt}=A(x-y)+R$? Dec8 comment Proof of a theorem about oscillation Since the first derivatives of $x$ and $y$ are given so we can write a Taylor series expansion of $x(t)$ and $y(t)$. It gives a result of the form $x(t)-y(t)= x_0^2 h(t,x_0)$ for some function $h$. If one can show that $h$ is bounded in a rectangular region $[0,T]\times[-a,a]$ with some upper bound $M$, then the theorem will follow. Since then $|x(t)-y(t)|<\delta \epsilon$ if we take $|x_0|<\delta$ and $\delta<\epsilon/M$ Aug31 comment Probability for the sum of two random numbers being a prime number? If the choice of numbers is in the range $N_1$ to $N_2$ (where both $N_1$ and $N_2$ are 'large') then the probability will be roughly $N_2/((N_2-N_1)log(N_2))−N_1/((N_2-N_1)log(N_1))$. Right ? Jun28 comment why symplectic form should be closed when we work on a manifold Now when you are asking for an example I think my statement is not correct;) It should be other way around - every symplectic vector space is also a symplect manifold. Doesn't the condition $d\omega=0$ follow from bilinearity? Since matrix elements of $\omega$ will have no dependence on coordinates. Jun28 comment why symplectic form should be closed when we work on a manifold I guess that "symplectic vector space" and "symplectic manifold" are two different notions. In particular not every symplectic vector space is a symplectic manifold. Jul23 comment Projective representations of loop groups But from Lie algebra point of view study of projective representations will also give you ordinary ones. Right ? Jul23 comment Projective representations of loop groups I am not sure. If there is any mathematical reason I too would like to know :) Jul12 comment basis functions do not lie in the space they form In case your question was inspired by quantum mechanics you may find this wikipedia page useful. Jul10 comment For which topological spaces $X$ can one write $X \approx Y \times Y$? Is $Y$ unique? thanks for your answer and references