(i) Consider the infinite series (v) Hence show that 1 1 = Bo, 0 = +...

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(i) Consider the infinite series (v) Hence show that 1 1 = Bo, 0 = + n = Bn Bn Bo+Bız + =x² B₂ 2! ex By expanding the LHS and equating powers of a determine that Во B₁ 2! 1! Show that for n > 1 one can write the relations as (B+1)"B" = 0 where BS → Bg. Show that generalising x to complex variable z (analytic continuation) n! 2 dz mi $c e²-1 2041 Σπί where ((s) = evaluate (2) and ((4) Bn = and 0 where the contour C encircles the origin in an anticlockwise fashion with |z| < 2 to avoid the poles at ±2πi. Locate and classify all the singularities of the integrand. (iv) By the residue theorem Bn = 2πi x residue from origin. By considering instead a large circle of radius R→ ∞ encircling the origin, show that B₁2πi x sum of all other residues. = 0 for n odd (-1) ¹/2n! (2π)" + = -2. . B₁ n! n Во B₁ B₂ + 3! 2! 1! + 1! 2! (n) for n even Em-1 ms. By calculating explicitly the Bernouilli numbers in (i), (i) Consider the infinite series (v) Hence show that 1 1 = Bo, 0 = + n = Bn Bn Bo+Bız + =x² B₂ 2! ex By expanding the LHS and equating powers of a determine that Во B₁ 2! 1! Show that for n > 1 one can write the relations as (B+1)"B" = 0 where BS → Bg. Show that generalising x to complex variable z (analytic continuation) n! 2 dz mi $c e²-1 2041 Σπί where ((s) = evaluate (2) and ((4) Bn = and 0 where the contour C encircles the origin in an anticlockwise fashion with |z| < 2 to avoid the poles at ±2πi. Locate and classify all the singularities of the integrand. (iv) By the residue theorem Bn = 2πi x residue from origin. By considering instead a large circle of radius R→ ∞ encircling the origin, show that B₁2πi x sum of all other residues. = 0 for n odd (-1) ¹/2n! (2π)" + = -2. . B₁ n! n Во B₁ B₂ + 3! 2! 1! + 1! 2! (n) for n even Em-1 ms. By calculating explicitly the Bernouilli numbers in (i),

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i Expanding the lefthand side LHS of the given series using the Taylor series expansion of ex we hav... View the full answer