Question

Q1 2016

a) We want to develop a method for calculating the function f(x) = $\int_{0}^{x}$sin(t)/t dt

for small or moderately small values of x. this is a special function called the sine integral, and it is related to another special function called the exponential integral. it rises in diffraction problems.

Derive a Taylor-series expression for f(x), and give an upper bound for the error when the series is terminated after the n-th order term. sint = see image

b)we need to evaluate the function h(x) for very large values of x, then matlab is used, it gives the following values (see image)

Here, Inf anf Nan stand for infinity and Not a Number. Explain what has happened, and show how to rewrite the function h(x) to fix this.

c) How should the fraction be written, so that it can be evaluated accurately for large x?

1. (a) We want to develop a method for calculating the function sin f dt f(x) 0 for small or moderately small values of x. This is a special function called the "sine integral", and it is related to another special function called the "exponential integral". It arises in diffraction problems Derive a Taylor-series expression for f(x), and give an upper bound for the when the series is terminated after the n-th order term. error [HINT +1.2n- (-1)/,-1 (2j-1) (-)" j+1,2j-1 t sint= t- + 3!5! +R +...+ + (2n 1) jl where the remainder term is (2n) Sin R, (t) (2n)! 0

Answer 1

la) Sintdt t Shnt =t-4+5 +614142n-) + Rn 2n-) +Rn j (2j-1)! Sst 2jT n Sint jLii-) fe t Cjt22-1 j-(2j-Daj- tylon seresfr 1+ Ynt2n 2 4n As m as (ta