Abstract
Along the way, we will meet the dominated convergence theorem, and we also compute the Laplace transform of the Bessel function of order zero. It is one thing to derive these remarkable formulas and quite another to discover them. We begin by rederiving the Leibnitz series for π by an alternate method to that in §3.7.
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Notes
- 1.
These functions need not be continuous, they may be arbitrary .
- 2.
The general version is in Chapter 6
- 3.
This series converges for x > 0 by the Leibnitz test.
- 4.
Via the unit circle map \(x' + iy' = (\sqrt{a}x + i\sqrt{b}y)/(\sqrt{a}x - i\sqrt{b}y)\).
- 5.
More accurately, the normalized elementary symmetric polynomials
- 6.
\(\int _{0}^{\infty }g(t)\,dt = I^{2} +\sec ^{2}b\).
- 7.
x n ′ also depends on t.
- 8.
B(x, y) is finite by (5.4.7).
- 9.
\(\int _{0}^{\infty }g(t)\,dt =\varGamma (a)\varGamma (b) + C_{\epsilon }\varGamma (a + b)\).
- 10.
The integral is \(C\sqrt{2\pi /(-f''(c)-\epsilon )}\).
- 11.
By Exercise 5.2.14
- 12.
This identity is simply a reflection of the fact that every natural has a unique binary expansion (§1.6).
- 13.
By the monotone convergence theorem for series.
- 14.
In fact, below we see β = 1 and the radius is 2π.
- 15.
With \(f_{n}(x) = 1/(n^{2} + x)\) and \(g_{n} = r!/n^{2}\), \(\vert f_{n}(x)\vert + \vert f'_{n}(x)\vert +\ldots +\vert f_{n}^{(r-1)}(x)\vert \leq g_{n}\) and \(\sum g_{n} = r!\zeta (2)\).
- 16.
The index notation in \(\theta _{0}\), \(\theta _{+}\), \(\theta _{-}\) is not standard.
- 17.
This equality and its derivation are valid whether or not there are infinitely many primes.
- 18.
By replacing χ ± by the characters χ of the group (Z∕a Z)∗.
- 19.
Is light composed of particles or waves?
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Hijab, O. (2016). Applications. In: Introduction to Calculus and Classical Analysis. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-28400-2_5
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DOI: https://doi.org/10.1007/978-3-319-28400-2_5
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