Luboš Motl
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 Jun10 answered How to find the triangle matrix of a given matrix? Jun10 revised Is fourier series of a function with $e^{j\theta}$ replaced with a complex variable $z$ holomorphic on the unit disc? added 587 characters in body; added 97 characters in body; added 118 characters in body Jun10 answered Is fourier series of a function with $e^{j\theta}$ replaced with a complex variable $z$ holomorphic on the unit disc? Jun10 comment Finding power series representation of $\int_0^{\frac{\pi }{2}} \frac{1}{\sqrt {1 - k^2\sin^2{x}}}\;{dx}$ Sorry, @Asaf, where did you get the factor of $1/3$? What does it mean? Jun10 comment Finding power series representation of $\int_0^{\frac{\pi }{2}} \frac{1}{\sqrt {1 - k^2\sin^2{x}}}\;{dx}$ Be careful, if it is an exam problem and they will mechanically compare the result with a wrong official template, they may declare your correct answer incorrect and you will have to defend yourself - for which, I believe, you have all the weapons. Jun10 comment Question about direct sum of function space Dear @vonjd, your problem already starts with ${\mathbb R}^3$: the first sentence says that $V$ itself are functions from $U$ which is a subset of ${\mathbb R}^3$. $V$ itself are functions that take value in ${\mathbb R}$ and $V\oplus V\oplus V$ are functions from ${\mathbb R}^3$ to another ${\mathbb R}^3$. But functions from ${\mathbb R}$ never appear in your problem at all so I don't understand in what sense it would be "natural". Real numbers and their 3rd power are equally natural but only the latter appear in your problem. The tripling only affects the value of the function not the domain Jun10 awarded Enlightened Jun10 comment Finding power series representation of $\int_0^{\frac{\pi }{2}} \frac{1}{\sqrt {1 - k^2\sin^2{x}}}\;{dx}$ It's just a small digit "2" that someone missed, but you didn't. Maybe they thought it was a mark for a footnote. ;-) Jun10 revised Finding power series representation of $\int_0^{\frac{\pi }{2}} \frac{1}{\sqrt {1 - k^2\sin^2{x}}}\;{dx}$ added 329 characters in body; added 1 characters in body Jun10 comment Finding power series representation of $\int_0^{\frac{\pi }{2}} \frac{1}{\sqrt {1 - k^2\sin^2{x}}}\;{dx}$ Oh, I eventually noticed that it's actually the same thing. In that case, I am sure that your result is correct. In the original formula, the whole big ratio in the parentheses should be squared, otherwise it's correct. Jun10 awarded Nice Answer Jun10 answered Finding power series representation of $\int_0^{\frac{\pi }{2}} \frac{1}{\sqrt {1 - k^2\sin^2{x}}}\;{dx}$ Jun9 comment A radius for which the inverse of a function is well defined (Inverse Function Theorem) The four derivatives $\partial f_{1,2} / \partial (x,y)$ are all nonzero at $(1/2,0)$ and the matrix is non-singular over there, so it's clear that in a vicinity of the point, the inverse exists. Jun9 comment A radius for which the inverse of a function is well defined (Inverse Function Theorem) Maybe they just want an approximate formula for the inverse? I can't imagine how the formulae could simplify. Or maybe you're allowed to use some symbols inside? Something that makes the solution "non-explicit"? Jun9 answered Question about direct sum of function space Jun9 answered A radius for which the inverse of a function is well defined (Inverse Function Theorem) Jun9 comment A radius for which the inverse of a function is well defined (Inverse Function Theorem) I see, thanks, it's a comma. ;-) Jun9 comment A radius for which the inverse of a function is well defined (Inverse Function Theorem) Sorry, I may be missing something, but how can you invert a function of two variables? Your function is from $R^2$ to $R$, so it cannot be a simple function, so it can't be (fully) inverted. The inverse function would have to be from $R$ to $R^2$, right? Jun9 comment Finding number of matrices whose square is the identity matrix Hi, every $9\times 9$ matrix $A$ may be brought into the standard form $A=CDC^{-1}$ for a $D$ which is either diagonal or has the Jordan blocks on the diagonal. That's a basic result in algebra. In this form, $A^2 = CDC^{-1}CDC^{-1} = CD^2 C^{-1}$. It should be equal to $I = CC^{-1}$ which implies $D^2=I$. So $D$ has to have $\pm 1$ eigenvalue and one may check that the nondiagonal Jordan blocks would fail to produce $D^2=1$, too. Jun9 revised Finding number of matrices whose square is the identity matrix added 263 characters in body