Page 619 - Handbook Of Integral Equations

For the chosen branch, the cut is a curve of discontinuity. On the edges of the cut we have

+
–
ln(z – z 0 ) = ln(z – z 0 )+ i2π,
–
γ
+
γ
(z – z 0 ) = e i2πγ (z – z 0 ) .
This discontinuity property of branches of multivalued functions on the edges of a cut is widely
used in the solution of boundary value problems with discontinuous boundary conditions. The
logarithm is applied for the case in which a discontinuous function enters the boundary condition as
a summand, and the power function corresponds to the case of a discontinuous factor in the boundary
conditions.

12.2-5. The Principal Value of a Singular Curvilinear Integral
Let L be a smooth contour and let τ and t be complex coordinates of its points. Consider the singular
curvilinear integral

ϕ(τ)
dτ. (11)
τ – t
L
Let us take a circle of some radius ρ centered at the point t on the contour. Let t 1 and t 2 be the
points of intersection of this circle with the curve. Assume that the radius is so small that the circle
has no other points of intersection with L. Let l be the part of the contour L cut out by the circle.
Consider the integral over the remaining arc,

ϕ(τ)
dτ. (12)
τ – t
L–l
The limit of the integral (12) as ρ → 0 is called the principal value of the singular integral (11).
Using the representation
ϕ(τ) ϕ(τ) – ϕ(t) dτ

dτ = dτ + ϕ(t)
τ – t τ – t τ – t
L L L
and the same reasoning as above, we see that the singular integral (11) exists in the sense of the
Cauchy principal value for any function ϕ(τ) satisfying the H¨ older condition.
At any point of smoothness, this integral can be presented in two forms:
ϕ(τ) ϕ(τ) – ϕ(t) b – t

dτ = dτ + ϕ(t) ln + iπ
τ – t τ – t a – t
L L
ϕ(τ) ϕ(τ) – ϕ(t) b – t

dτ = dτ + ϕ(t)ln ,
τ – t τ – t t – a
L L
where a and b are the endpoints of L.
In particular, if the contour is closed, then by setting a = b we obtain

ϕ(τ) ϕ(τ) – ϕ(t)
dτ = dτ + iπϕ(t).
τ – t τ – t
L L
Throughout the following, any singular integral will be understood in the sense of the Cauchy
principal value.
Let L be a smooth contour (closed or nonclosed) and let ϕ(τ)beaH¨ older function of a point on
the contour. Then the Cauchy type integral

1 ϕ(τ)
Φ(z)= dτ (13)
2πi τ – z
L

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 602