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 Jan22 awarded Popular Question Jan13 comment Compute complex Gaussian integral I can only show the nonvanishing near $z=0$, when $z$ is large, certainly there are zeros of $\det(A(z))$. Jan12 comment Compute complex Gaussian integral It seem in order to show that $f(z)=\sqrt{\det A(z)}$, where $A(z)=A+z(e_{ij} +e_{ji} )$, is holomorphic, I need to show that $C∖\{\det(A(z))|Re(A(z))>0\}$ can be cutted by a path from $0$ to $\infty$ , but what I can show is only that $f(z)$ is nonvanishing near $z=0$. Jan11 comment Compute complex Gaussian integral It seems not easy to show that both sides are holomorphic, in fact use the complex one variable theory, it is not clear what is the domain space of $z_{ij}=a_{ij}+b_{ij}i$. Jan11 accepted Compute complex Gaussian integral Jan10 comment Compute complex Gaussian integral @paulgarrett can you formulate an answer? Jan10 revised Compute complex Gaussian integral added 45 characters in body Jan10 comment Compute complex Gaussian integral @NickThompson Sorry I forgot a condition, that $A$ is symmetric. Thanks Paul garrett, I check the excercise again, and find the missing condition. Jan10 comment Compute complex Gaussian integral @NickThompson Yes. Jan10 revised Compute complex Gaussian integral added 246 characters in body Jan10 revised Compute complex Gaussian integral added 246 characters in body Jan10 comment Compute complex Gaussian integral It means $A=B+C i$, $B$ is positive definite, i.e., all its eigenvalue is positive. Jan10 asked Compute complex Gaussian integral Dec28 comment Existence of first order PDEs? Thanks for your interesting in this question, and make user7530's idea more clear, I also thanks JLA provided an explicit formula, and here angin as you have point out, the independence of path is infact is the closedness of the 1-form $\omega$ (plus Stokes theorem). What's more, your have prompt an really interesting quesiton, in what manner the bondary conditions of PDEs integrate into this geometric view? Dec28 comment Existence of first order PDEs? @user7530 I think the boundary condition only needed when we require the uniquness, at least this should be true, if the compitable condition is holds in $\bar\Omega$. Dec28 asked Existence of first order PDEs? Dec28 comment exactly represent Dirac distribution as derivatives of continuous function @MichaelAlbanese Thanks, it is caused by the post button not working properly when I try to submit my question. Maybe any someone can merge this two post? Dec26 comment Represent Dirac distruibution as a combination of derivatives of continuous functions? @paulgarrett Any reason or reference for this( $|x|'$ is not a function anymore)? Dec26 comment Represent Dirac distruibution as a combination of derivatives of continuous functions? @paulgarrett In fact, I am a litter confused in calculating $f_2(x)=|x|$'s distribution derivative, why $f_2'(0)=0$? Dec26 comment Represent Dirac distruibution as a combination of derivatives of continuous functions? @paulgarrett exactly!