Series Representation
A Taylor series for Bring radicals, as well as a representation in terms of hypergeometric functions can be derived as follows. The equation can be rewritten as ; by setting, the desired solution is .
The series for can then be obtained by reversion of the Taylor series for (which is simply ), giving:
where the absolute values of the coefficients are sequence A002294 in the OEIS. The series confirms that is odd. This gives
The series converges for |z| < 1 and can be analytically continued in the complex plane. The above result can be written in hypergeometric form as:
Compare with the hypergeometric functions that arise in Glasser's derivation and the method of differential resolvents below.
Read more about this topic: Bring Radical
Famous quotes containing the word series:
“A sophistical rhetorician, inebriated with the exuberance of his own verbosity, and gifted with an egotistical imagination that can at all times command an interminable and inconsistent series of arguments to malign an opponent and to glorify himself.”
—Benjamin Disraeli (1804–1881)