Consider next the integral where a, b 0. Do as much as you can from the tasks be

Question

Consider next the integral where a, b 0. Do as much as you can from the tasks below. a Find the location of the saddle point ro in terms of a and b b) By examining the second and possibly higher derivatives of the argument of the exponential at the saddle point, and by considering the ar-factor multiplying the exponential, determine the conditions on a and b that should be met in order to make the saddle-point approximation a good approximation (c) Evaluate the integral above employing the saddle point approximation, while using T ro for the factor multiplying the exponential. Would using a aro (a ro) produce a different result in the saddle-point approximation? d For a 1 and b 5, compute I numerically using a calculator or the Wolfram Alpha site (http:// If the upper limit of infinity cannot be used, take a large value for the upper limit, such as arma 20. Move the upper limit up and down, to make sure that the numerical integral has converged Compare the result from the saddle-point approximation above to the numerical result. What error in percentage is made using the saddle-point approximation? Note: Do not expect any high accuracy here. (e) Discuss how the accuracy of the saddle-point approximation could be improved for the specific calculation above by inclusion of higher order terms under the integral. Indicate what those terms would be. You do not need to carry out an explicit calculation for those terms

Explanation / Answer

Saddle point for a function is a stationary point such that the curve or surface in the neighbourhood of that point is not entirely contained on any side of the tangent space of that point.

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