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.
Contents
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
Corollary of Poisson integral formula
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
,
|
z
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