**III. Boundary Value Problems of Function Theory.
Their Basic Concepts**

We have already said that the theory of functions of a complex variable and especially its geometric part - the theory of conformal mapping - arose and was developed on the basis physical ideas. Leonhard Euler and Jean d'Alembert came to the analyticity of such functions from hydro-mechanical work. Bernhard Riemann in his studies constantly used interpretations of analytic functions, linked to plane flow of fluids and heat flow.

On the other hand, conversely, the development of the theory of functiosn of a complex variable allowed to extablish new methods of solution of important practical problems from diiferent sections of mathematical sciences ( hydro- and aero-dynamics, elasticity theory, electro-statics, magnetic and hear flows, etc.). We should note that a lot of the relevant work was done by Russian scientists.

Niklai Egorovich Joukowski and Serbei Aleksandrovich Chaplygin (1869 - 1942) obtained many of the basic results when applying function theory to hydro- and aero-dynamics. The methods of the theory of functions of a complex variable had a very important role in their important papers as well as in Joukowsky's book "The Theoretical Foundations of Aeronautics" (1911). Also in the second part of the 20-th Century, important applications of function theory to hydro- and aero-dynamics were published by M.V.Keldysh, S.A.Christiianovich, V.V. Golubev, L.I. Sedov, et.al.

G.V.Kolosov (1867 - 1936) initiated in 1909 the use of function theory in the plane theory of elasticity. The brilliant solution of these problems by methods, based on function theory, was found in the Twenties by N.I.Muskhelishvili. He explained these methods in his book [10], which appeared first in 1932 and was translated by Rainer Radok into English in 1951. The methods of the function theory also occupy a large part in thudies of different fields of physics (V.A.Fok, G.A.Hildert, et. al.).

We will consider in this chapter the basic physical ideas, linked to the theory of functions of a complex variable and its simplest applications. We start with the theory of harmonic functions of two variables, closely linked to the potentials of plane vector fields, the basic boundary value problems of the theory of harmonic and analytic functions and then, on the basis of the developed theory, concern ouraselves with the basic questions of applications.

*Functions, harmonic in the domain D *are
called the real functions *u*(*x*, *y*) of
the two real variables, which have in this domain continuous real
continuous second second order partial derivatives and satisfy
the differential equation*

( is the symbol of the
differential operator). As a rule, this equation is called *Laplace's
equation*. However, Laplace considered it only in 1782, while
Euler used earlier it in his work on hydrodynamics and other
branches of mathematical physics. We note immediately that, since
the equation is linear, any linar combination

of harmonic functions *u*_{k}(*x*,
*y*) with real constant coefficients* a*_{k }is
again a harmonic function.

* We will always be here concerned
with harmonic functions of two variables, because they are
closely linked to analytic functions. In applications, harmonic
functions of three variables *u*(*x*, *y*,
*z*) are just as important; they satisfy the equation

which, however, we will not deal with.

As we will see in thsi chapter, the potentials of the very
important vector fields considered in Physics are harmonic
functions and any harmonic function may be represented physically
as the potential of a vector field. Hence, also in the general
case, harmonic functions are often called *potentials *and
the theory of harmonic functions *potential theory.*

**41.*** ***Properties
of harmonic functions **First of all, we will explain the
links between the concepts of analytic and harmonic functions.
They are expressed in the following two theorems:

**Theorem 1 ***The real and
imaginary parts of an arbitrary function* *f*(*z*)
= *u*(*x*, *y*) + *iv*(*x*, *y*)*
which is single-vakued and analytic in a domain D are in this
domain harmonic functions.*

The proof follows directly from the D'Alembert-Euler conditions

In fact, since analytic functions
have derivatives of all orders, Equation (1) my be differentiated
with respect to *x* and* y. *Differentiating the
first of these with resepct to *x*, the second with
respect to *y *and employing the theorem of the equality
of mixed derivatives, we find

whence

The proof for the function *v*(*x*,
*y*) is analogous.

Two functions *u*(*x*,
*y*) + *iv*(*x*, *y*), linked by the
D'Alembert-Euler conditions, harmonic in the domain *D*,*
*are said to be *conjygate*.

**Theorem 2 ***For
any function u*(*x*, *y*),* harmonic in a
simply-connected domain D one may find a harmonic function v*(*x*,
*y*) *conjugate with it.*

In fact, we consider the integral

where *z*_{0} = *x*_{0}
+ *iy*_{0}* *is a fixed and *z* = *x*
+ *iy *is a variable point in the domain *D*. By
Laplace's equation,

this integral does not depend on the path of integration and
is a function of *z* only; we denote this function by *v*_{0}(*x*, *y*). Using
the properties of curvi-linear integrals

(we may take the integral from *z*
to *z + h *along the horizontal segment, on which *dy*
= 0); analogously, we have Hence, *v*_{0}(*x*,
*y*) is also a known function, the conjugate to the
function *u*(*x*, *y*). Since the function
is determined exactly by its partial derivatives apart froma
constant term, the set of all harmonic functions, conjugate to *u*(*x*,
*y*), is given by

where *C *is an arbitary
(real) constant.

We note that in a
multiply-connected domain *D* Integral (2)

is, generally speaking, a *multi-valued*
function. It may assume different values along two paths *L *and
joining
the points *z*_{0} and *z*, if it is
impossible to deform these paths from one into the other without
leaving the domain *D*(i.e., if inside the domain, bouneed
by *L *and there are points which do not belong to *D*).
Obviously, we can follow in this case the corresponding reasoning
in **13.**** **and
can assert that in multiply-connected domains the general formula
for the values of the function *v*(*x*, *y*),
determined by (2), has the form

where *N*_{k}
are arbitrary integers and *G*_{k}*
*are integrals along closed contours *g*_{k},
each of which contains inside it one connected part of the
boundary of *D*:

(cf. Equations (2) and (3) in **13.**),
The constants *G*_{k }are called the *periods *of
the integral (2) or its *cyclic constants*.

If one may single out in a certain
domain *D', *lying in *D*, a single-valued and
continuous branch of the function *v*(*x*, *y*),
determined by (3), then this branch is obviously a harmonic
function, conjugate to *u*(*x*, *y*), whence
the function *v*(*x*, *y*) is a *multi-valued
harmonic function.* Note that the partial derivatives of thsi
function are single-valued:

this follows from (3).

Obviously, Theorem 2 may be reformualted as follows:

**Theorem 3' ***One
may consider any function, harmonic in D, as teh real or
imaginary part of some analytic function f*(*z*); it
is determined apart from a constant term which is imaginary or
real, respectively.

We will not exclude the case of
multiply-connected domains, whence the analytic function *f*(*z*)
may turn out to be multi-valued.

**Example **A
calculation of its partial derivatives shows that the function

is harmonic in the ring 0 < |*z*|
< ¥. Integral (2) has the
form

and represents in the ring 0 <
|*z*| < ¥ an
infinitely-valued function. The corresponding analytic function

is also infinitely-valued.

**Theorem 3 ***Any harmonic
function u*(*x*, *y*) *is an analytic
function of its arguments x and y, i.e., in the neighbourhood of
every point z*_{0}* = x*_{0}* + iy*_{0}*
of the domain D it can be represented in the form of the sum of
an aboslutely converging series*

In fact, *by Theorem *2', *u*(*x*,
*y*) can be viewed as the real part of the function *f*(*z*),
single-valued and analytic in some neighbourhood |*z - z*_{0}|
< *R *of the point *z*_{0}. Let in this
neighbourhood

where *c*_{n}* = **a*_{n}* *+ *i**b*_{n}. The absolute value of the

real part of the general term of Series (6)

does not exceed

and since by Abel's Theorem of **19.****
**it converges absolutely in any circle |*z - *z_{0}|
£ *r < R*, i.e., the series
converges
for *r < R, *also the series with the general term (7)
will converge absolutely for |*x - x*_{0}| + |*y
- y*_{0}| < *R*. This series also is one
for *u*(*x, y*). After regrouping its terms (which
is allowed due to the proved absolute convergence), we obtain the
required series (5), and the theorem has been proved.

In particular, it follows from the proved theorem
that harmonic functions have partical drivatives of all orders.
It is readily proved that also harmonic functions have the last
property.follows (cf. Theorem 1 of **17.**)

Based on Theorem 3, one may obtain practically
convenient methods for the establishment of an analytic function *f*(*z*)
from its known real part. Transforming in an elementary manner
the general term of Series(7) for *u*(*x*, *y*),
we obtain the representation of this function in the
neighbourhood of the point *z*_{0}:

:

By Abel's Theorem, this series converges also for
complex values *x* and *y*, sufficiently close to *x*_{0}
and *y*_{0}, whence we may set in it

where *z* is a
point sufficiently close to *z*_{0}, and we
obtain:

Replacing here *z *by
*z*, we find after simple transformations

Fomula (8) has been obtained for points *z*, close to *z*_{0},
but, by the Uniqueness Theorem, it obviously remains true in the
entire domain of definition of *f*(*z*), because in
thsi doamin both parts of (8) are analytic functions of *z*.

In particular, if *f*(*z*) is analytic at the
origin of co-ordinates, then one may set *z*_{0} =
0, and (8) assumes the specially simple form:

We now present several examples of the application of (8) and (9):

by (9);

by (8), *z*_{0} = 1;

by (8), *z*_{0} = *p*/2.

In these three formulae, *C* is a purely
imaginary constant.

We now will consider the properties of harmonic
functions. On the basis of Theroems 1 and 2, these properties are
easily obtained from the corresponding properties of analytic
functions. For the sake of convenience, we will write again *u*(*z*)
instead of *u*(*x*, *y*), *u*(*P*)
instead of *u*(*x*_{1}, *x*_{2},
··· , *x*_{n}) for functions of
several variables, undersatnding by *P *the point with the
co-ordinates (*x*_{1}, *x*_{2},
··· , *x*_{n}).

**Theorem 4 **(mean value) *If
the function u*(*z*)* is continuous in the closed
circle with radius r with centre at the point z and harmonic
inside this circle, then*

The proof follows directly from (5) in **14.****
**by separation the of real part.

**Theorem 5 ***A harmonic
function other than a constant cnnot attain an extreme value at
an internal point of its domain of definition.*

It is sufficient to prove the theorem for the
case of a maximum, because the point of a minimum of a harmonic
function* u*(*z*) is a the point of the maximum of
the function -*u*(*z*) which is also harmonic.
Assuming the contrary, we we propose that the harmonic function *u*(*z*)
attains a maximum at the internal point *z*_{0} of
the domain.

We construct in the neighbourhood of the point *z*_{0}
a single-valued function *f*(*z*) such that *u*
= Re *f*(*z*). The function *e*^{f(z)}
is analytic and its modulus *e*^{u(z)},
by our assumption, attainsa maximum at an internal point *z*_{0}
of the domain. This contradicts the maximum principle in **15.****,**
whence the theorem has been proved.

One might prove Theorem 5 directly on the basis
of the mean value theorem just as the maximum principle has been
proved in **15.**

**Theorem 6 ***If the function
u(z), harmonic in the entire open plane is bounded, be it above
or below, it is constant.*

In fact, let *u*(*z*) be bounded
above: *u*(*z*) < *M. *We construct a
function *f*(*z*), analytic in the entire open
plane, such that *u*(*z*)=Re *f*(*z*).
By the condition of the theorem, all values of the function *w
= f*(*z*) lie in the half-plane *u* < *M*,
whence, by the Note at the end of **28.****,**
the function *f*(*z*) is constant, i.e., also *u*(*z*)
is constant.

The following two theorems establish the
character of *level lines *of harmonic functions, i.e.,
the sets of the points at which *u*(*z*) = const.

**Theorem 7 ***If not a
constant the harmonic function u*(*z*) has a closed
level line *u*(z) = *u*_{0}*, then
insode this line lies at least one special point* of this
function.*

** At such a point, the
function is not harmonic.*

In fact, otherwise *u*(z), continuous in a closed domain,
bounded by a level line, must attain a largest value *u*(z_{1})
and a smallest value *u*(z_{2}). By Theorem 5, the
points z_{1 }and z_{2 }must lie on the boundary
of the domain, i.e., on a level line; consequently, *u*(z_{1})
= *u*(z_{2}) and the function *u*(z) is
constant.

**Theorem 8 ***Any
sufficiently small neighbourhood of a point z*_{0 }*of
a level line u*(*z*)* = u*_{0 }*subdivides
this line into an even number* 2*n *(*n *³ 1) *of sections
in which* *u*(z)* assumes in turn values larger and
less than u*_{0 }*.*

The function *u*(*z*)*
- u*_{0 }vanishes at the point *u*_{0 };
the function *v*(*z*) becoming there equal to it so
that *v*(*z*_{0}) = 0, we obtain the
analytic function

which also vanishes at *z*_{0}.
We denote by *n* the order of this zero when we have in
the neighbourhood of the point *z*_{0}

and, consequently,

where z - *z*_{0 }=
*re*^{i}^{j},
*A *¹ 0, *B *some
constant and *o*(*r*^{n})
denotes a small quantity of order higher than *r*^{n}* *as *r *® 0. Hence, it is seen that for
sufficiently small *r*, as *j *ranges
from 0 = 2*p*, *u - **u*_{0 }vanishes 2*n*
times, changing irs sign, and the theorem has been proved.

In exactly the same way one proves that the level
lines of the harmonic function *v*(*z*), conjugate
to *u*(*z*), passing through the point *z*_{0 }, are subdivided in the
neighbourhood of thsi point into *n* branchestouching in *z*_{0
}the bisectors, mentioned in Theorem *.

It follows from Theorem 8
that the level lines of harmonic functions heve only simple
points (*n* = 1) or multiple points * with different
tangents (*n* > 1) - the case of isolated points, end
points or points of return are excluded.

* In any closed
domain, the harmonic property of functions *u*(*z*)
yield a finite number of multiple points of level lines (at each
such point *f*'(*z*)=0); other wise, by the
uniqueness theorem (**20.**), one must have *f '*(*z*) º 0.

In the sequel, it will be useful to note the following proposition - the inverse of the mean value theorem.

**Theorem 9 ***If
the function u*(*z*) *is continuous in the domain D
and at any point z for sufficiently small r*

*then the function u*(*z*)
is harmonic in D.

Our proof is based on the
existence theorem of harmonic functions, taking on the boundary
of a simply-connected domain given values; this theorem will be
proved in **43. **Let *z*_{0 }- an
arbitary point in *D* and _{0} - a closed
simply-connected domain which belongs to *D* and contains
the point *z*_{0}. By the existence theorem, we
construct the harmonic function *u*_{0}(*z*)
which assumes on the boundary *C*_{0}* *of _{0}
the same valueas also the function *u*(*z*) and
denote *U*(*z*) = *u*_{0}(*z*)
- *u*(*z*).

By the dsign and conditions
of the proved theorem, *U*(*z*) is continuous in _{0 }and
equal to zero on the boundary of this domain. Besides, the value
of *U*(*z*) at the centre of any circle, belonging
to _{0
}is equal to the arithemtic mean of its values on
circumefrence of this circle, because both the functions *u*(*z*)
and *u*_{0}(*z*) have this property: the
first, by assumption, the second by the mean value theorem.

Hence it follows that the
function *U*(*z*) cannot attain an extreme value at
internal points of *D*_{0}; the proof of thsi
proposition applies only to the continuity of the functions and
the mean value theorem (cf., the comments after Theorem 5).
However, since a function *U*(*z*), continuous in a
closed domain, must attain its extreme values, it attains them on
the boundary of _{0}. But, since on the boundary everywhere *U*(*z*)
= 0, then the largest and smallest values of *U*(*z*)
equal zero, whence *U*(*z*) º 0 everywhere on _{0}.
This means that everywhere in _{0 }the
function *u*(*z*) coincides with the jarmonic
function *u*_{0}(*z*) and, in particular,
is harmonic at the point *z*_{0}. Since *z*_{0
}is an arbitary point of *D*, the theorem has been
proved.

We proceed next to a
theorem, analogous to Weierstrass' Theorem in **19.**

**Theorem 10 ***Let
there be given a sequence of functions u*_{0}, *u*_{1},
··· , *u*_{n}, ···, *harmonic in
the domain D and continuous in **. If the series
converges uniformly on the boundary of D, then it also converges
uniformly inside D, where its sum is a function, harmonic in D.*

The uniform convergence of
the series inside *D *follows from the extremum principle.
In fact, by a known Cauchy cobvergence test *, there follows from
the uniform convergence of the series * on the boundary of the
domain D that for any **e **> *0
one can find an integer *N *such that for any* n > N
*any positive integer p and all points* **z** *of the boundary

Since the absolute sum of the terms here is harmonic, by the extremum principle, also for all points of the domain

Hence follows the uniform
convergence of the series *. *There remains to
prove that the sum of this series *u*(*z*) is a
ahrmonic fucntion. We use for this purpose Theorems 9 and 4. For
any sufficiently sma;; *r*, we have

(thsi integration of the
series is admissible due to its uniform convergence). By Theorem
3, the integrals on the right hand side are equal to 2*p**u*_{k}(*z*),
whence

and, by Theorem 9, the
function *u*(*z*) is harmonic at the point *z*.
The theorem has been proved, since *z* is an arbitary
point of the domain *D*.

In conclusion, we note yet two theroems, useful in the sequel. The first of them expresses that the property of fucntions to be harmonic is not violated by analytic transformation of the independent variable.