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A variation on the principle of mathematical induction is the so-called strong induction principle3. Formally, in a proof by strong induction, there is no separate basis step and the induction step uses the assumption ( V k : 0 ^ in order to prove P.n. The precise formulation is as follows. Principle of Strong Mathematical Induction. Let P be a predicate on the natural numbers. Then, with dummy n ranging over the natural numbers, (Vn::P.n) = < V n :: P.n <= {Vfc:0^ The word 'strong' is used because the induction hypothesis when proving P.(n+l) is P.O A P.I A ... A P.n, which is stronger than the hypothesis P.n used in simple mathematical induction. The absence of a base case is sometimes illusory; occasionally, a special case needs to be made for n = 0 in the induction step as the induction hypothesis is (V k: 0 < k < 0: P.k), i.e. true, which is just the same as in simple induction.

which when rearranged gives: The effect of these changes on the analysis carried out above occurs solely in the integral I4, which for the most general case would become:

1 76

If the particular 3D case were chosen, then this gives:

(4.6)

Example 12.1 1. Very often in the analysis of algorithms we are faced with a recurrence relation whose solution expresses, for example, the worst-case running time of the algorithm. An example is the following:

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Recent work has demonstrated the effect of such a central cell correction for this 3D case [109]. While the physical reasoning behind the addition of such a correction seems sound, here, as always, caution needs to be taken in choosing a value for A. Cynically, A could be looked upon as merely another parameter (although it is not a variational parameter), that can be varied in order to produce agreement between experiment and theory. The evaluation of I4 for the most general, variable symmetry case will be left as an exercise for the reader! 5.7 SPECIAL CONSIDERATIONS FOR ACCEPTORS Hitherto concentration has been focused on donor energies in heterostructures, although, as discussed earlier in this chapter, in principle all of the analysis above is also applicable to the calculation of acceptor levels. Agreement between the hydrogenie model of the acceptor in bulk and experiment needs to account for both the degeneracy of the valence band and the reduced screening of the negatively charged acceptor ion potential. In heterostructures however, the confinement potential lifts the degeneracy of light- and heavy-holes, so the only additional consideration that needs to be made relates to the permittivity. The procedure would be to deduce empirically the central cell correction parameter A, by comparing the calculated acceptor binding energy EAo predicted by the above

T.O = 1 ,

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