with respect to z0 .
Proof: We know from Thm. 10 that we have a contour integral representation
of f at any point z0 U :
f (z0 ) =
1
2i
f (z )
dz.
z z0
(758)
C
Choose a point z0 U . The derivative of f at z0 is given by
f (z0 + z ) f (z0 )
,
z 0
z
f (z0 )
18.6
Review of the Second Part of the Lecture
18.6.1
Final Exam
The nal exam of MATH 529 will take place on Tuesday, May 3, 2011, 8:00
11:00 AM, in Phillips Hall, Room 332 (the regular lecture hall). Please
read the paragraph on Final Examinations in the
12.14
Review of Chapter 12
12.14.1
Basic Concepts (12.1)
A partial dierential equation of order k N for an unknown function u :
R, Rd open, d > 1 (the total number of variables), is an expression
of the form
F (Dk u)(x), (Dk1 u)(x), . . . , (Du)(x), u(x)
18.3
Heat Problems
Laplaces equation also governs steady heat problems (cf. Sections 12.5, 12.6):
the heat equation is given by Tt = c2 T , where T [K] denotes the temperature and where c [m2 s1 ] denotes the thermal diusivity. We have Tt 0 in
steady stat
Therefore, the tangent mapping is a linear fraction transformation, preceded by an exponential mapping and followed by a clockwise rotation
by 90 .
With Z = e2y e2ix , we see that the strip S := cfw_z C | x (/4, /4)
in the z -plane is mapped onto the righ
17.2
Linear Fractional Transformations
An important class of mappings are linear fraction transformations (or Mbius
o
transformations), which are of the form
w = f (z ) =
az + b
,
cz + d
(990)
a, b, c, d C, ad bc = 0. Then we have
f (z ) =
a(cz + d) (az +
17
Conformal Mapping
We now go back to the main road, which leads from Chapter 12 (PDEs),
via Chapters 13, 14 & 17, to Chapter 18 (Potential Theory in 2D).
In this chapter, we interpret complex functions f : C C as mappings
from R2 to R2 , and thus consid
Remarks:
The function f (z ) = tan(1/z ) has a nonisolated singularity at z0 = 0.
We will not discuss this type of singularity here.
f has an essential singularity z0 C if and only if the principal part of
the Laurent series of f at z0 contains innitely
so that the general solutions are given by (Proposition 1)
Gmn (t) = Bmn cos(mn t) + Bmn sin(mn t),
Bmn , Bmn R,
(271)
for m, n N. The functions umn := Fmn Gmn ,
m
n
x sin
y,
a
b
(272)
m, n N, are eigenfunctions of the second-order dierential operators in
12.7
2D Wave Equation: Vibrating Membrane
In a similar way as for the vibrating string (Section 12.2) we now want
to derive a mathematical model of small transverse vibrations of an elastic
membrane, such as a drumhead. The membrane at rest covers a bound
12.6
1D Heat Equation: Solution by Fourier Integrals
and Transforms
In the previous section, we stated the one-dimensional heat equation as a
model for the heat conduction in a thin bar of length L > 0, which is laterally
insulated. We may now consider th
Note to myself: redraw 1D wave equation characteristics with correct slopes.
12.5
Heat Equation: Solution by Fourier Series
Mathematical model Consider a body of homogeneous material and denote its temperature by u(x, t) [K], x R3 , t 0. Let V denote a
bo
12.4
DAlemberts Solution of the Wave Equation. Characteristics.
Recall the series solution of the wave equation (61) with homogeneous Dirichlet boundary conditions (62) (value of u prescribed as 0) and initial data f ,
g:
u(x, t) =
Bn cos
n=1
n
n
ct + Bn
12.3
1D Wave Equation: Separation of Variables, Use
of Fourier Series
In the previous section, we have derived the one-dimensional wave equation
(60), which governs small transverse vibrations of an elastic string of length
L > 0. This PDE is given in :=
Note: Thursday oce hours changed to 2:00 3:30 PM.
12
Partial Dierential Equations (PDEs)
As mentioned in the overview, PDEs often come up in the mathematical
modeling of physical phenomena in continuous media, such as the propagation of sound, heat conduc