4/9/2017
CrosswordPuzzleMaker:FinalPuzzle
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Note:YoucanuseadifferentprogramnamedCrosswordWeavertoprintanice
1. See last exam notes
2.
Conservation biology understands human impacts on ecosystems, predicts future impacts,
changes unfavorable outcomes, restores species and habitats, and discovers new ways humans and
nature can coexist.
Research Cases
1. Invasive
definition, in rectangular form, in polar form,
and geometrically, using Cartesian coordinates
or polar coordinates. Each of these five ways is
useful in different situations, and translating
between them is an essential ingredient in
complex analysis. Th
to use the above definition to show that a set is
connected, since we have to rule out any
possible separation. CHAPTER 1. COMPLEX
NUMBERS 12 Example 1.10. The intervals X = [0,
1) and Y = (1, 2] on the real axis are separated:
There are infinitely many c
2.1. Use the definition of limit to show for any
zo C that limzz0 (az + b) = az0 + b. 2.2.
Evaluate the following limits or explain why
they dont exist. (a) lim zi iz31 z+i (b) lim
z1i (x + i(2x + y) 2.3. Prove that, if a limit
exists, then it is unique.
contributions along the new path will cancel
each other. The effect is that we transformed an
integral for which two singularities were inside
the integration path into a sum of two integrals,
each of which has CHAPTER 5. CONSEQUENCES
OF CAUCHYS THEOREM 7
< 1 (c) 0 < |z 1| < 2 (d) |z 1| + |z + 1| = 2 (e)
|z 1| + |z + 1| < 3 (f) |z| Re(z) + 1 1.28.
What are the boundaries of the sets in the
previous exercise? 1.29. Let G be the set of
points z C satisfying either z is real and 2 < z
< 1, or |z| < 1, or z =
6 t + i 2 (t 1) if 3 t 5 Figure 1.8: Two paths
and their parametrization. Figure 1.8 shows two
examples. We remark that each path comes
with an orientation, i.e., a sense of direction.
For example, the path 1 in Figure 1.8 is
different from 3(t) = 2 + 2 e
means that we can write any complex number
(x, y) as a linear combination of (1, 0) and (0, 1),
with the real coefficients x and y. Now (1, 0), in
turn, can be thought of as the real number 1. So
if we give (0, 1) a special name, say i, then the
complex n
using complex numbers, as was probably first
exemplified by Rafael Bombelli (15261572). In
the next section we show exactly how the
complex numbers are set up, and in the rest of
this chapter we will explore the properties of
the complex numbers. These pr
argument 2. This means we can write x1 + iy1
= (r1 cos 1) + i(r1 sin 1) and x2 + iy2 = (r2 cos
2) + i(r2 sin 2). To compute the product, we
make use of some classic trigonometric
identities: (x1 + iy1)(x2 + iy2) = (r1 cos 1 + i r1
sin 1) (r2 cos 2 + i r2
|z| 1 z 1.4. Play with other examples until you
get a feel for these functions. Chapter 2
Differentiation Mathematical study and
research are very suggestive of mountaineering.
Whymper made several efforts before he
climbed the Matterhorn in the 1860s and
your favorite online bookseller. About the
authors. Matthias Beck is a professor in the
Mathematics Department at San Francisco
State University. His research interests are in
geometric combinatorics and analytic number
theory. He is the author of two oth
approach a point z0 = (x0, y0) C. It is logical,
then, that there should be a relationship
between the complex derivative f 0 (z0) and the
partial derivatives f x (z0) := limxx0 f(x, y0)
f(x0, y0) x x0 and f y (z0) := lim yy0
f(x0, y) f(x0, y0) y y0 (so
something equivalent for the first term in (2.5),
since now both x and y are involved, and
both change as z 0. Instead we apply the
Mean-Value Theorem A.2 for real functions, to
the real and imaginary parts u(z) and v(z) of f(z).
Theorem A.2 gives real nu
to regions in which every closed path is
contractible. Definition. A region G C is
simply connected if G 0 for every closed
path in G. Loosely speaking, a region is simply
connected if it has no holes. Example 5.7. Any
disk D[a,r] is simply connected, as
for holomorphic functions, we will obtain such
a result much more easily; so we save the
derivation of integral formulas for higher
derivatives of f for later (Corollary 8.11).
Theorem 5.1 has several important
consequences. For starters, it can be used t
Abelian group with unit element (0, 0);
equations (1.10)(1.14) say that (C \ cfw_(0, 0), ) is
an Abelian group with unit element (1, 0). The
proof of Proposition 1.1 is straightforward but
nevertheless makes for good practice (Exercise
1.14). We give one
prove that it must be of the form z = e 2i a n
for some a R, then write a = m + b for some
integer m and some real number 0 b < 1, and
then argue that b has to be zero.) 1.18. Show
that z 5 1 = (z 1) z 2 + 2z cos 5 + 1 z 2 2z
cos 2 5 + 1 and deduce from t
D[a,r] is the circle C[a,r]. One notion that is
somewhat subtle in the complex domain is the
idea of connectedness. Intuitively, a set is
connected if it is in one piece. In R a set is
connected if and only if it is an interval, so there
is little reason
g 0 (z0) = limzz0 1 (g(z) = 1 limzz0 g(z)
= 1 f 0(w0) = 1 f 0(g(z0) . 2.3 Constant Functions
As a sample application of the definition of the
derivative of a complex function, we consider
functions that have a derivative of 0. In a typical
calculus course
semester undergraduate course developed at
Binghamton University (SUNY) and San
Francisco State University, and has been
adopted at several other institutions. For many
of our students, Complex Analysis is their first
rigorous analysis (if not mathematics
possible arguments for complex numbers, as
both cosine and sine are periodic functions with
period 2. 3Peter Hilton (Invited address,
Hudson River Undergraduate Mathematics
Conference 2000). 4 In particular, while our
notation proves Eulers formula e 2i =
:= ( 1 if Re z > 0, 2 if Re z < 0, then f 0 (z) = 0 for
all z in the domain of f , but f is not constant.
This may seem like a silly example, but it
illustrates a pitfall to proving a function is
constant that we must be careful of. Recall that
a region o
oriented clockwise. Show that R f = R f even
though 6G where G = C \ cfw_1, the region
of holomorphicity of f . 4.38. This exercise gives
an alternative proof of Cauchys Integral
Formula (Theorem 4.27) that does not depend
on Cauchys Theorem (Theorem 4.18
exclude the origin z = 0 from the domain. On
the other hand, we could construct some
functions that make use of a certain
representation of z, for example, f(x, y) = x 2iy,
f(x, y) = y 2 ix, or f(r, ) = 2r ei(+) . Next we
define limits of a function. The
I, the function f must be constant on I. We do
not (yet) have a complex version of the MeanValue Theorem, and so we will use a different
argument to prove that a complex function
whose derivative is always 0 must be constant.
Our proof of Proposition 2.13