The Egorychev method is a collection of techniques introduced by Georgy Egorychev for finding identities among sums of binomial coefficients, Stirling numbers, Bernoulli numbers, Harmonic numbers, Catalan numbers and other combinatorial numbers. The method relies on two observations. First, many identities can be proved by extracting coefficients of generating functions. Second, many generating functions are convergent power series, and coefficient extraction can be done using the Cauchy residue theorem (usually this is done by integrating over a small circular contour enclosing the origin). The sought-for identity can now be found using manipulations of integrals. Some of these manipulations are not clear from the generating function perspective. For instance, the integrand is usually a rational function, and the sum of the residues of a rational function is zero, yielding a new expression for the original sum. The residue at infinity is particularly important in these considerations.
Some of the integrals employed by the Egorychev method are:
 formula_1
where formula_2
 formula_3
where formula_4
 formula_5
where formula_6
 formula_7
where formula_4
 formula_9
where formula_4
 formula_11
where formula_12
Example I.
Suppose we seek to evaluate
formula_13
which is claimed to be :formula_14
Introduce :formula_15
and :formula_16
This yields for the sum : 
formula_17
This is 
formula_18
Extracting the residue at formula_19 we get 
 formula_20
thus proving the claim.
Example II.
Suppose we seek to evaluate formula_21
Introduce
formula_22
Observe that this is zero when formula_23 so we may extend formula_24 to
infinity to obtain for the sum
 formula_25
Now put formula_26 so that (observe that with formula_27 the image of formula_28 with formula_29 small is another closed circle-like contour which makes one turn and which we may certainly deform to obtain another circle formula_30)
formula_31
and furthermore
formula_32
to get for the integral
formula_33
This evaluates by inspection to (use the Newton binomial)
 formula_34
Here the mapping from formula_35 to formula_19 determines
the choice of square root. For the conditions on formula_37
and formula_38 we have that for the series to converge we
require formula_39 or formula_40 or formula_41 The closest that the image
contour of formula_42 comes to the origin is
formula_43 so we choose formula_44 for example formula_45 This also ensures that formula_46 so formula_30 does not intersect the branch
cut formula_48 (and is contained in the image of
formula_42). For example formula_50
and formula_51 will work.
This example also yields to simpler methods but was included here to demonstrate the effect of substituting into the variable of integration.
Computation using formal power series.
We may use the change of variables rule 1.8 (5) from the Egorychev text
(page 16) on the integral
formula_52
with formula_53 and formula_54 We
get formula_55 and find
formula_56
with formula_57 the inverse of formula_58. 
This becomes
formula_59
or alternatively
formula_60
Observe that formula_61
so this is
formula_62
and the rest of the computation continues as before.