- Source: Schwarz integral formula
In complex analysis, a branch of mathematics, the Schwarz integral formula, named after Hermann Schwarz, allows one to recover a holomorphic function, up to an imaginary constant, from the boundary values of its real part.
Unit disc
Let f be a function holomorphic on the closed unit disc {z ∈ C | |z| ≤ 1}. Then
f
(
z
)
=
1
2
π
i
∮
|
ζ
|
=
1
ζ
+
z
ζ
−
z
Re
(
f
(
ζ
)
)
d
ζ
ζ
+
i
Im
(
f
(
0
)
)
{\displaystyle f(z)={\frac {1}{2\pi i}}\oint _{|\zeta |=1}{\frac {\zeta +z}{\zeta -z}}\operatorname {Re} (f(\zeta ))\,{\frac {d\zeta }{\zeta }}+i\operatorname {Im} (f(0))}
for all |z| < 1.
Upper half-plane
Let f be a function holomorphic on the closed upper half-plane {z ∈ C | Im(z) ≥ 0} such that, for some α > 0, |zα f(z)| is bounded on the closed upper half-plane. Then
f
(
z
)
=
1
π
i
∫
−
∞
∞
u
(
ζ
,
0
)
ζ
−
z
d
ζ
=
1
π
i
∫
−
∞
∞
Re
(
f
)
(
ζ
+
0
i
)
ζ
−
z
d
ζ
{\displaystyle f(z)={\frac {1}{\pi i}}\int _{-\infty }^{\infty }{\frac {u(\zeta ,0)}{\zeta -z}}\,d\zeta ={\frac {1}{\pi i}}\int _{-\infty }^{\infty }{\frac {\operatorname {Re} (f)(\zeta +0i)}{\zeta -z}}\,d\zeta }
for all Im(z) > 0.
Note that, as compared to the version on the unit disc, this formula does not have an arbitrary constant added to the integral; this is because the additional decay condition makes the conditions for this formula more stringent.
The formula follows from Poisson integral formula applied to u:
u
(
z
)
=
1
2
π
∫
0
2
π
u
(
e
i
ψ
)
Re
e
i
ψ
+
z
e
i
ψ
−
z
d
ψ
for
|
z
|
<
1.
{\displaystyle u(z)={\frac {1}{2\pi }}\int _{0}^{2\pi }u(e^{i\psi })\operatorname {Re} {e^{i\psi }+z \over e^{i\psi }-z}\,d\psi \qquad {\text{for }}|z|<1.}
By means of conformal maps, the formula can be generalized to any simply connected open set.
Notes and references
Ahlfors, Lars V. (1979), Complex Analysis, Third Edition, McGraw-Hill, ISBN 0-07-085008-9
Remmert, Reinhold (1990), Theory of Complex Functions, Second Edition, Springer, ISBN 0-387-97195-5
Saff, E. B., and A. D. Snider (1993), Fundamentals of Complex Analysis for Mathematics, Science, and Engineering, Second Edition, Prentice Hall, ISBN 0-13-327461-6
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