ABSTRACT
The integral in Eq. (1) can also be written in a frequently used equivalent form. To this end, we consider the following transformation of the kernel:
M (t, τ ) τ – t
=
M (t, τ ) – M (t, t) τ – t
+ M (t, t) τ – t
, (2)
where we set M (t, t) = b(t), 1
πi
M (t, τ ) – M (t, t) τ – t
= K(t, τ ). (3) In this case Eq. (1), with regard to (2) and (3), becomes
a(t)ϕ(t) + b(t) πi
∫ L
ϕ(τ ) τ – t
dτ +
∫ L
K(t, τ )ϕ(τ ) dτ = f (t). (4)
It follows from formulas (3) that the function b(t) satisfies the Ho¨lder condition on the entire contourL andK(t, τ ) satisfies the Ho¨lder condition everywhere except for the points τ = t, at which one has the estimate
|K(t, τ )| < A|τ – t|λ , A = const <∞, 0 ≤ λ < 1.
