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44 1. Mathematical preliminaries

Q1.14 An integral I. Find the first four terms in an asymptotic expansion of the integral

for (a) (b)
Q1.15 An integral II. Find the first four terms in an asymptotic expansion of the integral

for (a) (b) (and in this case, introduce the constants

Q1.16 An integral III. Find the first two terms in an asymptotic expansion of the integral

as Explain why your result is not valid if [Hint: simply
use integration by parts.]
Q1.17 An integral IV. Use an appropriate integration by parts to find an expansion,
valid as of the integral

in particular, find the general term and also an expression for the remainder.
Show that this is an asymptotic expansion and that the series diverges. Find an
estimate for the remainder and use this to select the number of terms, for a
given x, which will minimise the error when using the series to find the value
of I (x).
Q1.18 Approximation for a Bessel function. The Bessel function is a solution of the
equation

first write find the equation for z(x) and show that, for large
x, this becomes approximately. Hence seek a solution