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Physics / An Introduction to Thermal Physics 1 / Chapter B / Problem 19P

An Introduction to Thermal Physics | 1st Edition | ISBN: 9780201380279 | Authors: Daniel V. Schroeder

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Integrate equation B.42 twice more, then plug in x = ? /2

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QUESTION:

Problem 19P

Integrate equation B.42 twice more, then plug in x = π /2 to obtain a formula for Σodd(1/k4). Use this formula to show that ζ(4) = π4/90 and thus evaluate the integrals B.36 for the case n = 3. Explain why this procedure does not yield a value for ζ(3).

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QUESTION:

Problem 19P

Integrate equation B.42 twice more, then plug in x = π /2 to obtain a formula for Σodd(1/k4). Use this formula to show that ζ(4) = π4/90 and thus evaluate the integrals B.36 for the case n = 3. Explain why this procedure does not yield a value for ζ(3).

ANSWER:

Step 1

Equation B.42 is given by

Now we have to integrate the above equation twice and then we have to use the value of the integration to calculate the value of $\zeta{}(4)$ .

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