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

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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
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
Page 39: The Casimir Effect Making Sense of
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
Page 89 and 90: Zeta Function Regularisation Regula
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Zeta Function Regularisation Regula
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