94
9 Plane Problem of Elasticity Theory
) +
1
σ
−
1
σ
1
σ
−
1
σ 2
1
σ
= f (θ),
(9.28)
where
f (θ) = f 1 (θ ) + if 2 (θ ) = N + iT .
If we switch to conjugate values in formula (9.28), we obtain
1
σ
+ (σ ) − σ σ
(σ ) − σ
2 (σ ) = f (θ).
(9.29)
By multiplying both formulas (9.28) by
1
2πi
dσ
σ − ζ
and integrating along the outline of the single circumference γ , we obtain
1
2πi
⎡
⎣
γ
(σ )
σ − ζ
dσ +
γ
1
σ
dσ
σ − ζ
−
γ
1
σ
dσ
σ (σ − ζ )
−
γ
1
σ
dσ
(σ − ζ )σ 2
⎤
⎦ =
1
2πi
γ
f dσ
σ − ζ
.
By using integral formulas of Cauchy and taking into account that
1
ζ
is a
holomorphic function beyond γ , we have from the last formula
) =
1
2πi
γ
(N + iT )dσ
σ − ζ
− a 0 ,
(9.30)
where
a 0 = (0) =
1
4πi
γ
(N − iT )
dσ
σ
.
(9.31)
Acting in a similar way with the condition (9.29), we obtain
(0) + ) − ζ ζ
(ζ ) − ζ
2 (ζ ) =
1
2πi
γ
f (θ)dσ
σ − ζ
.
(9.32)