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Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 8

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 2 part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Result 13.7.1 Integrals from Zero to Infinity. Let f z be a single-valued analytic function with only isolated singularities and no singularities on the positive real axis 0 to . Let a ị Z. If the integrals exist then I f x dx -V Res f z log z zk 0 k 1 i2n 2 x f x dx 1 e 2n 2 Res z f z zk f x log x dx -1 Res f z log2 z zk in Res f z log z zk 0 2 k i k i x f x Iogxdx 2J2 a zRes z f z logz zk 1 - e k i 2 n XL X Res zaf z zk sin na t 1 dm i2n xf x log x dx da 1 - 12 Res z f z zk where z1 . zn are the singularities of f z and there is a branch cut on the positive real axis with 0 arg z 2n. 654 13.8 Exploiting Symmetry We have already used symmetry of the integrand to evaluate certain integrals. For f x an even function we were able to evaluate 0 f x dx by extending the range of integration from X to X. For i xaf x dx Jo we put a branch cut on the positive real axis and noted that the value of the integrand below the branch cut is a constant multiple of the value of the function above the branch cut. This enabled us to evaluate the real integral with contour integration. In this section we will use other kinds of symmetry to evaluate integrals. We will discover that periodicity of the integrand will produce this symmetry. 13.8.1 Wedge Contours We note that zn rn eznd is periodic in 0 with period 2n n. The real and imaginary parts of zn are odd periodic in 0 with period n n. This observation suggests that certain integrals on the positive real axis may be evaluated by closing the path of integration with a wedge contour. Example 13.8.1 Consider r 1 T - dx Jo 1 xn .