Download Review Sheet for Analytic and Entire Functions | MATH 3912 and more Study notes Mathematics in PDF only on Docsity!
Analytic and Entire Functions
Mercedes Lueck
March 4, 2004
Contents
1 What is a Complex Function? 1
2 Limits and Continuity of Complex Functions 2
3 Complex Differentiation 3
4 Analyticity 5
5 Entire 7
1 What is a Complex Function?
We shall begin with a review of the basics.
Definition 1.1 (Function) A function f is a rule that assigns to each element in a set A one and only one element in a set B.
If f assigns the value b to the element a in A, we write
b = f (a)
and call b the image of a under f. As a result of this definition of function we sometimes refer to f as a mapping of A onto B. We are not concerned with just any functions. We are concerned with com- plex valued funtions of a complex variable, that is z = x + ıy, where x is the real part and y is the imaginary part (both real-valued). For complex valued functions, if w denotes the value of the function f at the point z, we then write
w = f (z).
Also, just as the complex number z decomposes into real and imaginary parts, the complex function w also decomposes into real and imaginary parts, again both real-valued. This is customarily written
w = u(x, y) + ıv(x, y),
with u and v denoting the real and imaginary parts respectively. Thus, a complex-valued function of a complex variable is, in essence, a pair of real functions of two real variables.
Example 1.1 Write the function w = z^2 +2z in the form w = u(x, y)+ıv(x, y).
Solution: By setting z = x + ıy we obtain
w = (x + ıy)^2 + 2(x + ıy) = x^2 − y^2 + ı 2 xy + 2x + ı 2 y. Which then can be rewritten as
w = (x^2 − y^2 + 2x) + ı(2xy + 2y).
2 Limits and Continuity of Complex Functions
The concepts of limits and continuity for complex functions are similar to those for real functions. Let’s first examine the concept of the limit of a complex- valued function.
Definition 2.1 (Limit) Let f be a function defined in some neighborhood of z 0 , with the possible exception of the point z 0 itself. We say that the limit of f (z) as z approaches z 0 is the number w 0 and write
lim z→z 0 f (z) = w 0 ,
or equivalently,
f (z) → w 0 as z → z 0 , if for any > 0 there exists a positive number δ such that |f (z) − w 0 | < whenever 0 < |z − z 0 | < δ.
This is esentially the same definition the we learned in calculus I when we were first learning limits. The condition of continuity is as follows.
Definition 2.2 (Continuous) Let f be a function defined in a neighborhood of z 0. Then f is continuous at z 0 if
lim z→z 0 f (z) = f (z 0 ).
3 Complex Differentiation
In general, a complex function of a complex variable, f (z), is an arbitrary mapping from the xy-plane to the uv -plane. A complex function is split into real and imaginary parts, u and v, and any pair u(x, y) and v(x, y) of two- variable functions gives us a complex function u + ıv. However, notice there is something special about the pair
u 1 (x, y) = x^2 − y^2 , and v 1 (x, y) = 2xy, as opposed to u 2 (x, y) = x^2 − y^2 , and v 2 (x, y) = 3xy. The difference is that the complex function u 1 + ıv 1 treats z = x + ıy as a single ”unit”, because x^2 − y^2 + ı 2 xy = (x + ıy)^2. These are the types of functions that are complex differentiable.
Definition 3.1 Let f be a complex-valued function defined in a neighborhood of z 0. Then the derivative of f at z 0 is given by
df dz
(z 0 ) ≡ f ′(z 0 ) := lim ∆z→ 0
f (z 0 + ∆z) − f (z 0 ) ∆z
provided this limit exists. (Such an f is said to be differentiable at z 0 ).
The catch in this definition is that ∆z is a complex number, so it can ap- proach zero in many different ways. Even though this catch may make things seem slightly more difficult, the rules for differentiating real functions apply in the same way for complex-valued functions (as long as the complex-valued function is in a form where z = x + ıy is treated as a single unit).
Example 3.1 Show that, for any positive integer n,
d dz
zn^ = nzn−^1.
Solution: Using Definition 3.1 we have
(z + ∆z)n^ − zn ∆z
nzn−^1 ∆z + n(n 2 − 1)zn−^2 (∆z)^2 + · · · + (∆z)n ∆z
Thus d dz
zn^ = lim ∆z→ 0
[
nzn−^1 +
n(n − 1) 2
zn−^2 ∆z + · · · + (∆z)n−^1
]
= nzn−^1.
Theorem 3.1 If f and g are differentiable at z, then
(f ± g)′(z) = f ′(z) ± g′(z), (1) (cf )′(z) = cf ′(z) for any constant c, (2) (f g)′(z) = f (z)g′(z) + f ′(z)g(z), (3) ( f g
(z) =
g(z)f ′(z) − f (z)g′(z) g(z)^2
if g(z) 6 = 0. (4)
(5)
If g is differentiable at z and f is differentiable at g(z), then the chain rule holds: d dz
f (g(z)) = f ′(g(z))g′(z).
4 Analyticity
Now that we have a secure background we are ready to look at the theory of analytic functions.
Definition 4.1 A complex-valued function f (z) is said to be analytic on an open set G if it has a derivative at every point of G.
Analyticity is a property defined over open sets, while differentiability could hold at one point only. If the phrase ”f (z) is analytic at the point z 0 ” is used it means that f (z) is analytic in some neighborhood of z 0. To show that a function is analytic we use the following equations which must hold at z 0 = x 0 + ıy 0 :
∂u ∂x
∂v ∂y
∂u ∂y
∂v ∂x known as the Cauchy-Riemann equations.
Theorem 4.1 A necessary condition for a function f (z) = u(x, y) + ıv(x, y) to be differentiable at a point z 0 is that the Cauchy-Riemann equations hold at z 0. Consequently, if f is analytic in an open set G, then the Cauchy-Riemann equations must hold at every point of G.
Example 4.1 Show that the function f (z) = (x^2 + y) + ı(y^2 − x) is not analytic at any point.
Solution: Since u(x, y) = x^2 + y and v(x, y) = y^2 − x we have ∂u ∂x = 2x,^
∂v ∂y = 2y, ∂u ∂y = 1,^
∂v ∂x =^ −1.
Example 4.4 Find
C
ez (z+5)^3 (z−ı) dz, where^ C^ is the circle centered at the origin of radius 2.
Solution: Again in this example we must rewrite the problem in the form f (z) z−z 0.
C
ez (z + 5)^3 (z − ı)
dz
C
ez (z+5)^3 (z − ı)
dz
In this case it is easy to see that f (z) = e
z (z+5)^3 and^ z^0 =^ ı.^ In this case z 0 = ı is included in C so Theorem 4.3 can be applied.
C
ez (z+5)^3 (z − ı)
dz
= 2 πıf (ı)
= 2 πı
eı (ı + 5)^3
5 Entire
If f (z) is analytic on the whole complex plane, then it is said to be entire.
Definition 5.1 A function f (z) is called entire if it has a representation of the form
f (z) =
∑^ ∞
k=
akzk^ valid for |z| < ∞.
This class of functions is designated by E. E is a linear space.
Example 5.1 Some examples of entire functions are
sin(z) z
, 2 z^ ,
∫ (^) z
0
et
2 dt,
Γ(z)
Theorem 5.1 The function f (z) =
k=0 akz k (^) is entire if and only if
lim n→∞ |an| n^1 = 0.
