# T.A.E.

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Trial And Error.

A portrait of a famous Mathematician who was often criticized for not being rigorous enough in his new inventions.

"Anyone who has never made a mistake has never tried anything new." - Albert Einstein

 12 Proving that if $|f''(x)| \le A$ then $|f'(x)| \le A/2$ 10 Improper integral of $\int_0^\infty \frac{e^{-ax} - e^{-bx}}{x}\ dx$ 9 Show that $\int_0^{2\pi}\frac{R^2-r^2}{R^2 - 2Rr\cos (\varphi-\vartheta) + r^2}d\vartheta$ is independent of $R>r>0$, using only real numbers. 7 If a linear operator has an adjoint operator, it is bounded 7 Complex analysis is more “real” than real analysis

# 15,107 Reputation

 +10 Selfadjoint Restrictions of Legendre Operator $-\frac{d}{dx}(1-x^{2})\frac{d}{dx}$ +35 Associated Legendre polynomials +10 Spectrum: Polynomials +10 Difference between an eigenvalue and a spectral value

# 16 Questions

 5 How to show $e^{-x}$ is a cyclic vector for $-\frac{d^{2}}{dx^{2}}$ in $L^{2}[0,\infty)$? 5 Simple proof that $\|p(A)\|\le \sup_{|z|\le 1}|p(z)|$ for polynomials $p$ and $\|A\| \le 1$. 5 Show that $\|e^{tA}\| \le e^{t\|\Re (A)\|}$ 4 Proving Stone's Formula for Constructively obtaining the Spectral Measure for $A=A^\star$ 4 Is there a cyclic vector for $-\frac{d^{2}}{dx^{2}}$ on $L^{2}[0,2\pi]$ with periodic conditions?

# 197 Tags

 466 functional-analysis × 397 116 linear-algebra × 110 206 real-analysis × 139 89 analysis × 71 185 operator-theory × 154 78 complex-analysis × 55 134 spectral-theory × 89 72 calculus × 47 117 hilbert-spaces × 96 60 differential-equations × 76

# 3 Accounts

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