**for**

**11.3 Poisson's Integral
Formula for the Upper Half-Plane**

The Dirichlet problem for the upper half-plane is to find a function that is harmonic in the upper half plane and has the boundary values , where , is a real-valued function of the real variable x. An important method for solving this problem is our next result which is attributed to the French mathematician Siméon Poisson.

**Theorem 11.3 (**__Poisson's
Integral
Formula__**).** Let
U be a real-valued function that is
piecewise continuous and bounded for all real t. The
function

(11-12)

is harmonic in the upper half plane
and has the boundary values

wherever
is continuous.

**Proof of Theorem 11.3 is in the book.
**

**Example 11.11.** Find
the function
that is harmonic in the upper half-plane ,
which takes on the boundary values

Solution. Using Equation (11-12), we
obtain

Using techniques from calculus we have the integration
formula . We
obtain the solution as follows

**Extra Example
1.** Find the function
that is harmonic in the upper half-plane ,
which takes on the boundary values

**Example 11.12.** Find
the function
that is harmonic in the upper half-plane ,
which takes on the boundary values

Solution. Using Equation
(11-12), we obtain

Using techniques from calculus we have the integration formulas

, and .

We obtain the solution as follows

The function
is continuous in the upper half-plane, and on the boundary ,
except at the discontinuities
on the real axis. The graph in Figure 11.14 shows this
phenomenon.

The graph of with the boundary valuesFigure 11.14

**Example 11.13.** Use
Poisson's Integral formula to find the harmonic function
that is harmonic in the upper half-plane ,
that takes on the boundary values

Solution. Using techniques from Section
11.2, we find that the function

is harmonic in the upper half-plane and has the boundary
values This
function can be added to the one in Example 11.12 to obtain the
desired result:

Figure 11.15 shows the graph of .

The graph of .Figure 11.15

**Extra Example
2.** Use Poisson's Integral formula to find the
harmonic function
that is harmonic in the upper half-plane ,
that takes on the boundary values

**Exercises
for Section 11.3. Poisson's Integral Formula for the Upper
Half Plane**** **

**A Companion Module
is**

**Dirichlet
Problem for the Disk**

**The Next Module
is**

**Two-Dimensional
Mathematical Models**

**Return to the Complex
Analysis Modules**

__Return
to the Complex Analysis Project__

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2012 John H. Mathews, Russell W. Howell