We start with the subtracted singularities method. Observe that in the last example formula 8. We now re-derive the above formula using the contour integral approach. But by Cauchy's bound 8. We can summarize our discussion in the following theorem.

## Real-argument incomplete hankel functions

A binary string is a d; k sequence if it does not contain any runs of 0's shorter than d and longer than k. In other words, between any pair of 1's there must be at least d and at most k zeros. We want to enumeratePall such d; k sequences. We apply the symbolic methods of Chapter 7. Let Ad;k be the set of all words strings consisting only of 0's whose length is between d and k.

Thus, 8. The function Wd;k z is rational and we can compute jWd;k j exactly. Then the denumerator in 8.

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Let us consider a particular example. It is known that the function is multi-valued. These two functions are known as branches. Observe the following phenomenon: when we start from a given point z and choose one of the branches, say w1 , then when traversing any closed contour we come back to w1 , unless we enclose the origin.

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If the contour encloses the origin, then w attains the value w2 after one circle, and it comes back to w1 after the second loop. Such a point is called a branch point. We shall call it an algebraic singularity. This way we prevent z to encircle p the origin and thus avoid ambiguity.

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Naturally, z is not an exception. The reader may review this material from any standard textbook on complex analysis e. If, however, after a continuation p along a closed path the function does not return to its original branch as in the case of z , then the function is multi-valued and z0 is a branch point. The idea of the proof of Puiseux's expansion is quite simple. We next discuss in some depth how to obtain precise 2 2 asymptotic expansions for such functions around its algebraic singularities.

There are many techniques to deal with such singularities. However, in analysis of algorithms the singularity analysis of Flajolet and Odlyzko is the most useful, and we devote some time to it in the sequel. That's what we will do next. We shall follow Flajolet and Odlyzko []. Actually, because of these excellent references, we focus here on explaining things rather than on detailed proofs.

We will prove it using Hankel's contour that we review in Table 8.

Figure 8. We denote the integral along such loop as or 0. To see why 8. Thus, we prove 8. By analytic continuation since both sides of 8. Observe that if we traverse the Hankel loop in the reverse order i. We give a detailed proof of 8. Table 8.

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Using 8. To justify it rigorously, we must prove convergence of above series in order to interchange the integral and the sum. This is necessary in order to justify 8.

This completes the proof of 8. The proof of 8. We only sketch it. This completes the sketch of the proof for 8. For details the reader is referred to []. We are lucky if we get an expansion of A z around its singularities. In order to evaluate an as n! These transfer theorems are jewels of Flajolet and Odlyzko singularity analysis [], and we discuss them next. However, in order not to clutter too much our expositions, we consider a particular p example p adopted from Odlyzko [].

## Gamma function

From Theorem 8. Furthermore, this assumption will not allow us to not handle expression like 1 z log 1 1 z that often arises in analysis of algorithms. This is due to Darboux and we brie y discuss it at the end of this section. We now consider the third scenario analyzed by Flajolet and Odlyzko []. Finally, we are left with C2 and C4 which can be handled in a similar manner. Let us consider C2. We should observe that the critical derivation was the one on the rectilinear curves C2 and C4. We just established one of the transfer theorem of Flajolet and Odlyzko.

In summary, the singularity analysis of Flajolet and Odlyzko is based on the following transfer theorems. The proof follows the footsteps of the above derivations, and details can be found in []. Thus by 7.

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We shall discuss another example in the application Section 8. We still have some options. One is real analysis, another is Tauberian theorem. However, the most successful seems to be the Darboux method that we brie y discuss next. We start with a quick review of the Riemann-Lebesgue lemma that is often useful, as an intermediate result, in establishing asymptotics. Lemma 8. Then Z1 xlim! The proof of the Riemann-Lebesgue lemma can be found in [, ]. We just observe that it follows from a few simple facts.

In addition, since f t is continuous, it is bounded in [a; b]. Now, we present the Darboux method. In this case, we only assume that A z is smooth enough on the disk of convergence. Again, the proof follows from the Riemann-Lebesgue lemma. Finally, we should mention that so far we only dealt with a single singularity. Multiple singularities can be handled in a similar fashion, however, some complications arise. The reader is referred to Flajolet and Sedgewick [] for details. We recall that, roughly speaking, a singularity is large if the growth of the function near such a singularity is rapid, say, larger than any polynomial.

We start with the Laplace method which assumes that the path C in 8. Then, we discuss the saddle point method also known as the steepest descent method which by far is the most useful and popular technique to extract asymptotic information of rapidly growing functions. It is one of the most complicated method, therefore, given the purpose and limitations, we do not present a full discussion of it. The reader is referred to [67, , ] for a complete and insightful presentations. We complete this section with an application of the saddle point method to large deviations.

The reader is also referred to excellent survey by Odlyzko [] and the book by Flajolet and Sedgewick []. Observe then that as x! While above is only a heuristic, we will make it rigorous soon. We should add that the last line of the above derivation follows from the Gauss integral. The following lemma is often used in this book. The underlying principle, due to Lord Kelvin, is the assertion that the major asymptotic contribution comes from points where the phase of h t is stationary, that is, where h0 t vanishes.

We should add that oscillations near such points slow down. The last equation above follows from the Fresnel integral cf. We now focus on the Laplace method and prove the following theorem. We follow Lauwerier []. Combining everything we obtain 8.