1 1 Given an integer n 1, define J(n)=1+++ We want to extend the function...

 

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1 1 Given an integer n ≥ 1, define J(n)=1+++ We want to extend the function J to other values than positive integers. The starting point is the equality n 1+t+²+ +t" = 1-th 1-t 1-1+1 1- -dt. Hint: Rewrite the integrand as a finite sum (a) (3 marks) Show that J(n) = 1+1+... (b) (3 marks) Given a real number 20, define gr(t) = 1-tt 1-t Show that lim gr(t) = =x. Given any x ≥ 0, part (b) shows that gr(t) extends to a continuous function of t on [0, 1], so that the function J(x) == L 1 - tx 1-t is defined on [0, ∞) and, by part (a), coincides with J(n) on positive integers. (c) (3 marks) Given any x < 0, show that lim 9x (t) asymptote at t = 0. t→0+ -dt = -∞ so that gx (t) has a vertical (d) (5 marks) Show that for x ≥ 0, we have the recurrence relation J(x+1) - J(x) = = 1 x+1 (e) (6 marks) The recurrence relation enables us to extend the definition of J(x) to all real numbers except negative integers. Compute J(1/2), J(0), and J(-1/2). 1 1 Given an integer n ≥ 1, define J(n)=1+++ We want to extend the function J to other values than positive integers. The starting point is the equality n 1+t+²+ +t" = 1-th 1-t 1-1+1 1- -dt. Hint: Rewrite the integrand as a finite sum (a) (3 marks) Show that J(n) = 1+1+... (b) (3 marks) Given a real number 20, define gr(t) = 1-tt 1-t Show that lim gr(t) = =x. Given any x ≥ 0, part (b) shows that gr(t) extends to a continuous function of t on [0, 1], so that the function J(x) == L 1 - tx 1-t is defined on [0, ∞) and, by part (a), coincides with J(n) on positive integers. (c) (3 marks) Given any x < 0, show that lim 9x (t) asymptote at t = 0. t→0+ -dt = -∞ so that gx (t) has a vertical (d) (5 marks) Show that for x ≥ 0, we have the recurrence relation J(x+1) - J(x) = = 1 x+1 (e) (6 marks) The recurrence relation enables us to extend the definition of J(x) to all real numbers except negative integers. Compute J(1/2), J(0), and J(-1/2).