1. Sketch an antiderivative of the function ... Get solution
2. Take another derivative of equation to evaluate ... Equation:... Get solution
3. The integral of ...is easier to evaluate when n is odd.(a) Evaluate...(No computation allowed!)(b) Evaluate the indefinite integral (i.e., the antiderivative) of ... using a simple substitution.(c) Evaluate...(d) Differentiate the previous result to evaluate... Get solution
4. Sometimes you need to integrate only the “tail” of a Gaussian function, from some large x up to infinity:...Evaluate this integral approximately as follows. First, change variables to s = t2, to obtain a simple exponential times something proportional to s−1/2. The integral is dominated by the region near its lower limit, so it makes sense to expand s−1/2 Get solution
5. Use the methods of the previous problem to find an asymptotic expansion for the integral of ... from x to ∞, when x ≫ 1. Get solution
6. The antiderivative of...set equal to zero at x = 0 and multiplied by ...is called the error function, abbreviated erf x:...(a) Show that erf(±∞) = ±1.(b) Evaluate...in terms of erf x.(c) Use the result of Problem to find an approximate expression for erf x when x ≫ 1.Problem:Evaluate this integral approximately as follows. First, change variables to s = t2, to obtain a simple exponential times something proportional to s−1/2. The integral is dominated by the region near its lower limit, so it makes sense to expand s−1/2 Get solution
7. Prove the recursion formula. Do not assume that n is an integer.Fromula:... Get solution
8. Evaluate ... (Hint: Change variables to convert the integrand to a Gaussian.) Then use the recursion formula to evaluate ... Get solution
9. Carry out the integral numerically to evaluate ... and ... A useful identity whose proof is beyond the scope of this book is...Check this formula numerically for n = 1/ 3.Integral: ... Get solution
10. Choose the limits on the integral in equation more carefully, to derive a more accurate approximation to n!. (Hint: It’s the upper limit that is more critical. There’s no obvious best choice for the lower limit, but do the best you can.)Equation:... Get solution
11. Prove that the function xne−x reaches its maximum value at x = n. Get solution
12. Use a computer to plot the function xne−x and the Gaussian approximation to this function, for n = 10, 20, and 50. Notice how the relative width of the peak (compared to n) decreases as n increases, and how the Gaussian approximation becomes more accurate as n increases. If your computer software permits it, try looking at even higher values of n. Get solution
13. It is possible to improve Stirling’s approximation by keeping more terms in the expansion or the logarithm. The exponential of the new terms can then be expanded in a Taylor series to yield a polynomial in y multiplied by the same Gaussian as before. Carry out this procedure, consistently keeping all terms that will end up being smaller than the leading term by one power of n. (Since the Gaussian cuts off when y is of order ... you can estimate the sizes of various terms by setting ...) When the smoke clears, you should find...Check the accuracy of this formula for n = 1 and for n = 10. (In practice, the correction term is rarely needed. But it does provide a handy way to estimate the error in Stirling’s approximation.)Expand the logarithm:... Get solution
14. The proof of formula is by induction. (a) Check formula for n = 0 and for n = 1. (b) Show that...(Hint: First write (sin θ)n as (sin θ)n −2 (1 − cos2 θ). Integrate the second term by parts, differentiating one factor of cos θ and integrating everything else.)(c) Use the results of parts (a) and (b) to prove formula by induction.Formula:... Get solution
16. Derive a formula for the volume of a d-dimensional hypersphere. Get solution
17. Derive the general integration formulas B.36. Get solution
18. Use a computer to plot the sum of sine waves on the right-hand side of equation B.41, terminating the sum first at k = 1, then at k = 3, 5, 15, and 25. Notice how the series does converge to the square-wave function that we started with, but the convergence is not particularly fast. Get solution
19. 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). Get solution
20. Evaluate equation B.41 at x = π/2, to obtain a famous series for π. How many terms in this series must you evaluate to obtain π to three significant figures? Get solution
21. In calculating the heat capacity of a degenerate Fermi gas in Section 7.3, we needed the integral...To derive this result, first show that the integrand is an even function, so it suffices to integrate from 0 to ∞ and then multiply by 2. Then integrate by parts to relate this integral to the one in equation B.35. Get solution
22. Evaluate ζ(3) by numerically summing the series. How many terms do you need to keep to get an answer that is accurate to three significant figures? Get solution
2. Take another derivative of equation to evaluate ... Equation:... Get solution
3. The integral of ...is easier to evaluate when n is odd.(a) Evaluate...(No computation allowed!)(b) Evaluate the indefinite integral (i.e., the antiderivative) of ... using a simple substitution.(c) Evaluate...(d) Differentiate the previous result to evaluate... Get solution
4. Sometimes you need to integrate only the “tail” of a Gaussian function, from some large x up to infinity:...Evaluate this integral approximately as follows. First, change variables to s = t2, to obtain a simple exponential times something proportional to s−1/2. The integral is dominated by the region near its lower limit, so it makes sense to expand s−1/2 Get solution
5. Use the methods of the previous problem to find an asymptotic expansion for the integral of ... from x to ∞, when x ≫ 1. Get solution
6. The antiderivative of...set equal to zero at x = 0 and multiplied by ...is called the error function, abbreviated erf x:...(a) Show that erf(±∞) = ±1.(b) Evaluate...in terms of erf x.(c) Use the result of Problem to find an approximate expression for erf x when x ≫ 1.Problem:Evaluate this integral approximately as follows. First, change variables to s = t2, to obtain a simple exponential times something proportional to s−1/2. The integral is dominated by the region near its lower limit, so it makes sense to expand s−1/2 Get solution
7. Prove the recursion formula. Do not assume that n is an integer.Fromula:... Get solution
8. Evaluate ... (Hint: Change variables to convert the integrand to a Gaussian.) Then use the recursion formula to evaluate ... Get solution
9. Carry out the integral numerically to evaluate ... and ... A useful identity whose proof is beyond the scope of this book is...Check this formula numerically for n = 1/ 3.Integral: ... Get solution
10. Choose the limits on the integral in equation more carefully, to derive a more accurate approximation to n!. (Hint: It’s the upper limit that is more critical. There’s no obvious best choice for the lower limit, but do the best you can.)Equation:... Get solution
11. Prove that the function xne−x reaches its maximum value at x = n. Get solution
12. Use a computer to plot the function xne−x and the Gaussian approximation to this function, for n = 10, 20, and 50. Notice how the relative width of the peak (compared to n) decreases as n increases, and how the Gaussian approximation becomes more accurate as n increases. If your computer software permits it, try looking at even higher values of n. Get solution
13. It is possible to improve Stirling’s approximation by keeping more terms in the expansion or the logarithm. The exponential of the new terms can then be expanded in a Taylor series to yield a polynomial in y multiplied by the same Gaussian as before. Carry out this procedure, consistently keeping all terms that will end up being smaller than the leading term by one power of n. (Since the Gaussian cuts off when y is of order ... you can estimate the sizes of various terms by setting ...) When the smoke clears, you should find...Check the accuracy of this formula for n = 1 and for n = 10. (In practice, the correction term is rarely needed. But it does provide a handy way to estimate the error in Stirling’s approximation.)Expand the logarithm:... Get solution
14. The proof of formula is by induction. (a) Check formula for n = 0 and for n = 1. (b) Show that...(Hint: First write (sin θ)n as (sin θ)n −2 (1 − cos2 θ). Integrate the second term by parts, differentiating one factor of cos θ and integrating everything else.)(c) Use the results of parts (a) and (b) to prove formula by induction.Formula:... Get solution
16. Derive a formula for the volume of a d-dimensional hypersphere. Get solution
17. Derive the general integration formulas B.36. Get solution
18. Use a computer to plot the sum of sine waves on the right-hand side of equation B.41, terminating the sum first at k = 1, then at k = 3, 5, 15, and 25. Notice how the series does converge to the square-wave function that we started with, but the convergence is not particularly fast. Get solution
19. 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). Get solution
20. Evaluate equation B.41 at x = π/2, to obtain a famous series for π. How many terms in this series must you evaluate to obtain π to three significant figures? Get solution
21. In calculating the heat capacity of a degenerate Fermi gas in Section 7.3, we needed the integral...To derive this result, first show that the integrand is an even function, so it suffices to integrate from 0 to ∞ and then multiply by 2. Then integrate by parts to relate this integral to the one in equation B.35. Get solution
22. Evaluate ζ(3) by numerically summing the series. How many terms do you need to keep to get an answer that is accurate to three significant figures? Get solution
No hay comentarios:
Publicar un comentario