Tham khảo tài liệu 'mathematical method in science and engineering episode 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | PROBLEMS 473 where the degeneracy gn and the eigenfrequencies wn are given as Si 21 1 ít ẳì. Note Interested students can obtain the eigenfrequencies and the degeneracy by solving the wave equation for the massless conformal scalar field 7 t 0 4 n 1 where n is the dimension of spacetime R is the scalar curvature and is the d Alembert wave operator g dy where dy stands for the covariant derivative. Use the separation of variables method and impose the boundary condition 4 finite on the sphere. For this problem spacetime dimension n is 3 and for a sphere of constant radius Rq the curvature scalar is 2 fíg. Using asymptotic series evaluate the logarithmic integral Hint Use the substitutions t e and a In x a 0 and integrate by parts successively to write the series where Zoo -t so that . _ . n e. a SSr- In a closed Einstein universe the renormalized energy density of a massless conformal scalar field with thermal spectrum can be written as ren. 27T2 R fe _ 00 tic V n l exp n3 nhc hRịịT J he 240 n 474 INFINITE SERIES where Ro is the constant radius of the universe T is the temperature of the radiation and 2tt2 Ìq is the volume of the universe. The second term he inside the square brackets is the well-known renormalized quantum 240-rto vacuum energy that is the Casimir energy for the Einstein universe. First find the high and low temperature limits of p ren and then obtain the flat spacetime limit Ro oo. Without using a calculator evaluate the following sum to five decimal places How many terms did you have to add Check the convergence of the series S .1 2 q 2 4 f e Find the interval of convergence for the series ia y xn b y _ J n a 2- 1ln n 2 w ựĩi Evaluate the sums EE a cosnớ ồ En o n sin a is a constant Hint Try using complex variables. Verify the following Taylor series xn ex y for all X t- n 71 0 and Ỉ-X2 --- -iyix2n for x l. 1 X. PROBLEMS 475 Find the first three nonzero terms of the following Taylor series a f x X3
