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Using hypergeometric functions to determine the terminal speeds of parachutes

Summary

Engineering students are interested in the applications of science and mathematics. When mathematical subjects are presented to them as tools that help them solve problems, the relevance of those subjects and the very usefulness of mathematics increase in the eyes of the students. This paper illustrates the introduction of hypergeometric functions through their application in the determination of the terminal speeds of parachutes. It also demonstrates a simple way by which elementary functions such as logarithms and exponentials can be shown to relate to hypergeometric functions.

Introduction

A generalized hypergoemetric function denoted by p Fq (a1 ,... a p ; b1 ,...bq ; x ) can be defined using a hypergoemetric series. The function , which corresponds to , is known as Gauss’ hypergeometric function. It is regular and was the first one to be studied, perhaps, because it is encountered quite frequently in engineering and scientific applications 9,11. It is defined as

r ∞ ( a ) r ( b) r ( x k ) 2 F ([ a , b],[ c], x ) ≡ ∑ k , (1) 1 r=0 ( c) r r ! where ( a ) r is the Pochhammer symbol 4, for which Γ (a + r ) ( a ) r = a (a + 1)(a + 2)...(a + r − 1) = , (2) Γ (a )

and Γ denotes the gamma function given by the Euler Integral of the second kind 3.

Hypergeometric functions are solutions to the hypergeometric differential equation

z(1 − z) y ′′ + [c − (a + b + 1) z] y ′ − aby = 0 . (3)

Using the Froebenius method, the complete solution to this equation is shown to be 9

y = A2 F1 (a , b; c; z) + Bz1− c 2 F1 (a + 1 − c, b + 1 − c;2 − c; z ) (4)

The first two derivatives of 2 F1 (a , b; c; z ) are given, respectively, by 10

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Njock-Libii, J. (2006, June), Using Hypergeometric Functions To Determine The Terminal Speeds Of Parachutes Paper presented at 2006 Annual Conference & Exposition, Chicago, Illinois. 10.18260/1-2--33