Download An elementary treatise on elliptic functions by Arthur Cayley PDF

By Arthur Cayley

This quantity is made out of electronic photographs from the Cornell college Library ancient arithmetic Monographs assortment.

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6. Prove that the field of complex numbers cannot be made into an ordered field. (Hint: Since i = 0 then either i > 0 or i < 0. ) 7. Prove that the complex roots of a polynomial with real coefficients occur in complex conjugate pairs. 8. Calculate the square roots of i. 9. Prove that the set of all complex numbers is uncountable. 10. Prove that any nonzero complex number z has kth roots r1 , r2 , . . , rk . That is, prove that there are k of them. 11. In the complex plane, draw a picture of S = {z ∈ C : |z − 1| + |z + 1| = 2} .

3). However, just for the moment, use the definition you learned in calculus class and consider the sequence {sin j}∞ j=1 . Notice that the sequence is bounded in absolute value by 1. The Bolzano–Weierstrass theorem guarantees that there is a convergent subsequence, even though it would be very difficult to say precisely what that convergent subsequence is. 27 Let {αj } be a bounded sequence of complex numbers. Then there is a convergent subsequence. Proof: Write αj = aj + ibj , with aj , bj ∈ R.

2 Consider the series ∞ 2−j . j=1 The N th partial sum for this series is SN = 2−1 + 2−2 + · · · + 2−N . In order to determine whether the sequence {SN } has a limit, we rewrite SN as SN = 2−0 − 2−1 + 2−1 − 2−2 + . . 2−N +1 − 2−N . , successive pairs of terms cancel) and we find that SN = 2−0 − 2−N . Thus lim SN = 2−0 = 1 . N →∞ We conclude that the series converges. 3 Let us examine the series ∞ j=1 1 j for convergence or divergence. ) Now S1 = 1= S2 = 1+ S4 = 1+ ≥ 1+ S8 = 1+ ≥ 1+ = 5 .

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