Expanding the square roots in a binomial series shows that the spread in pumping frequencies that result in exponentially growing q is approximately \ omega _ { n } f _ { 0 }.
12.
Where the first term is the total relativistic energy, and the second term is the rest energy of the electron . ( c is the speed of light ) Expanding this in a Taylor series ( specifically a binomial series ), we find
13.
More examples of the use of the empty product in mathematics may be found in the binomial theorem ( which assumes and implies that " x " 0 = 1 for all " x " ), Stirling number, K�nig's theorem, binomial type, binomial series, difference operator and Pochhammer symbol.
14.
One guess I have is to take advantage of the integral definition of the inverse secant, \ arcsec ( x ) = \ int _ 1 ^ x \ frac { dt } { t \ sqrt { t ^ 2-1 } }, expand the square root with a binomial series, antidifferentiate, and then apply the Lagrange inversion theorem, but I do not have experience applying that.
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:Also recall that an analytic p-th root in the unit ball around the identity is provided by the binomial series, that works in any Banach algebra A . You may also like to investigate the set of all square roots of the identity : it is an analytic submanifold of A ( in general not connected ); the tangent space at " x " is given by the closed linear subspace V of all " v " that anti-commute with " x ", that is " vx =-xv ".
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