Analysis Fact

Daily tweets about real and complex analysis and related topics. From @JohnDCook.

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Applied mathematics consulting

https://t.co/AwpFbcBvIn

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Euler studied the gamma function in 1729, but Legendre first gave it the name 'gamma' in 1809.

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Solve for Jacobi sn parameter to have given period(s) https://t.co/GRwJU0vmda

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“As with any asymptotic technique, Laplace’s method — despite its simplicity — is replete with subtleties.” -- John Wojdylo

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Joseph Fourier introduced modern summation notation in 1820. (i.e. capital sigma with limits on bottom and top)

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sin(x - y) sin(x + y) = (sin(x) - sin(y)) (sin(x) + sin(y))

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Generalizing the fundamental theorems of calculus to measurable functions https://t.co/bgtegLtDPt

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Taylor series do not explain why

exp(−x²) ≈ (1 + cos(sin(x) + x))/2

is such a good approximation.

https://t.co/VOV6uZx7P5

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Another Gaussian approximation

https://t.co/7UbYyEePdd

Orders of magnitude better than an naive analysis would suggest

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If H(x) is a Hermite polynomial and phi(x)= exp(-x²/2)/√(2π) then H(x) phi(x) is an eigenfuction of the Fourier Transform.

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Applying the law of sines on a sphere https://t.co/jhp2TBa0bO

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All connected abelian Lie groups have the form T^n x R^m where T^n is an n-torus.

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For integers n >= 0 and m >= 1, the Bessel functions J_n(x) and J_{n+m}(x) have no common zeros except at x = 0.

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The Runge-Kutta method for numerically solving differential equations was developed in 1895.

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Twice lately I've assumed a problem doesn't have a nice solution, tried to solve it numerically, got stuck, then went back and found a simple analytical solution.

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Freshmen believe every problem has a simple closed-form solution, because that's true of every problem they've ever seen.

Real-world problems hardly ever have a closed-form solution, much less a simple one, but once in a while they do.

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Joukowsky airfoils https://t.co/34AYZ4OqEh

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A set K is a cone if tK is contained in K for all t >= 0.

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arctan(x) = x - x³/3 + x⁵/5 - ...

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Dmitri Mendeleev, best known for his periodic table, also discovered a polynomial inequality.

https://t.co/urb9eY6V26