The Mathematics of the Casimir Effect - School of Mathematical ...

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Divergent Series The Casimir Effect Making Sense of Infinity - Infinity Conclusion The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula On the Whole, Divergent Series are the Works of the Devil and it’s a Shame that one dares base any Demonstration upon them. You can get whatever result you want when you use them, and they have given rise to so many Disasters and so many Paradoxes. Can anything more horrible be conceived than to have the following oozing out at you: where n is an integer number? 0 = 1 − 2 n + 3 n − 4 n + etc. Niels Henrik Abel Thomas Prellberg The Mathematics of the Casimir Effect
The Casimir Effect Making Sense of Infinity - Infinity Conclusion Summing Divergent Series The Mathematical Setting Divergent Series Euler-Maclaurin Formula Abel-Plana Formula Some divergent series can be summed in a sensible way . . . S = ∞∑ (−1) n = 1 − 1 + 1 − 1 + 1 − 1 + . . . n=0 Cesaro summation: let S N = P N n=0 (−1)n and compute Abel summation: 1 S = lim N→∞ N + 1 S = lim x→1 − ∞ X n=0 NX S N = 1 2 n=0 (−1) n x n = 1 2 Borel summation, Euler summation, . . . : again S = 1 2 ∞∑ . . . but some (such as n) cannot n=0 Thomas Prellberg The Mathematics of the Casimir Effect
Page 1 and 2: The Casimir Effect Making Sense of
Page 5 and 6: Topic Outline The Casimir Effect Ma
Page 27 and 28: Spectral Theory The Casimir Effect
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Page 79 and 80: Outline The Casimir Effect Making S
Page 81 and 82: Conclusion The Casimir Effect Makin
Page 83 and 84: Conclusion The Casimir Effect Makin
Page 87 and 88: Zeta Function Regularisation Regula
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Zeta Function Regularisation Regula
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