Page 624 - Handbook Of Integral Equations

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1 . The index of a function that is continuous on a closed contour and vanishes nowhere is an integer
(possibly zero).
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2 . The index of the product of two functions is equal to the sum of the indexes of the factors. The
index of a ratio is equal to the difference of the indexes of the numerator and the denominator.
We now assume that D(t) is differentiable and is the boundary value of a function analytic in
the interior or exterior of L. In this case, the number

1 1 D (t)
t
ν = d ln D(t)= dt (4)
2πi L 2πi L D(t)
is equal to the logarithmic residue of the function D(t). The principle of argument (see Subsec-
tion 12.3-1) implies the following properties of the index:

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3 .If D(t) is the boundary value of a function analytic in the interior or exterior of the contour, then
its index is equal to the number of zeros inside the contour or minus the number of zeros outside the
contour, respectively.
4 . If a function D(z) is analytic in the interior of the contour except for finitely many points at
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which it may have poles, then the number of zeros must be replaced by the difference of the number
of zeros and the number of poles.
Here the zeros and the poles are counted according to their multiplicities. We also note that the
indexes of complex conjugate functions have opposite signs.
Let
t = t 1 (s)+ it 2 (s) (0 ≤ s ≤ l)

be the equation of the contour L. On substituting the expression of the complex coordinate t into
the function D(t), we obtain

D(t)= D t 1 (s)+ it 2 (s) = ξ(s)+ iη(s). (5)
Let us regard ξ and η as Cartesian coordinates. Then

ξ = ξ(s), η = η(s)

is a parametric equation of some curve Γ. Since the function D(t) is continuous and the contour L
is closed, it follows that the curve Γ is closed as well.
The number of turns of the curve Γ around the origin, i.e., the number of full rotations of the
radius vector as the variable s varies from 0 to l, is obviously the index of the function D(t). This
number is often called the winding number of the curve Γ with respect to the origin.
If the curve Γ is successfully constructed, then the winding number can be observed directly.
There are many examples for which the index can be found by analyzing the shape of the curve Γ.
For instance, if D(t) is a real or a pure imaginary function that does not vanish, then Γ is a line
segment (traversed an even number of times), and the index D(t) is equal to zero. If the real part ξ(s)
or the imaginary part η(s) preserves its sign, then the index is obviously zero, and so on. If the
function D(t) can be represented as the product or the ratio of functions that are limit values of
functions analytic in the interior or exterior of the contour, then the index can be calculated on the
basis of properties 2 ,3 and 4 .
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In the general case, the calculation of the index can be performed by formula (3). On the basis
of formula (5) we substitute the expression
η(s)
d arg D(t)= d arctan
ξ(s)

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 607