Chapter 2 Analytic Function 9. Functions of a complex variable Let S be a set of complex numbers. A function defined on S is a rule that assigns to each z in S a complex number w. or S is the domain of definition of sometimes refer to the function itself, for simplicity. Both a domain of definition and a rule are needed in order for a function to be well defined.

Suppose is the value of a function at real-valued functions of real variables x, y or Ex. when v=0 is a real-valued function of a complex variable.

is a polynomial of degree n. : rational function, defined when For multiple-valued functions : usually assign one to get single-valued function Ex.

w=f (z) is not easy to graph as real functions are. 10. Mappings w=f (z) is not easy to graph as real functions are. One can display some information about the function by indicating pairs of corresponding points z=(x,y) and w=(u,v). (draw z and w planes separately). When a function f is thought of in this way. it is often refried to as a mapping, or transformation. T z w image of z image of T inverse image of w

Mapping can be translation, rotation, reflection. In such cases it is convenient to consider z and w planes to be the same. w=z+1 translation +1 w=iz rotation w= reflection in real axis. Ex. image of curves real number y x a hyperbola is mapped in a one to one manner onto the line

right hand branch x>0, u=C1, left hand branch x<0 u=C1, V U V=C2>0 Y 2xy=C2 X image right hand branch x>0, u=C1, left hand branch x<0 u=C1,

a parabola with vertex at y D B A 1 L1 L2 C x1 Ex 2. u v A’ L2’ L1’ B’ D’ C’ When , point moves up a vertical half line, L1, as y increases from y = 0. a parabola with vertex at half line CD is mapped of half line C’D’

Ex 3. x y u v one to one

means: the limit of as z approaches z0 is w0 11. Limits Let a function f be defined at all points z in some deleted neighborhood of z0 means: the limit of as z approaches z0 is w0 can be made arbitrarily close to w0 if we choose the point z close enough to z0 but distinct from it. (1) means that, for each positive number ,there is a positive number such that (2) x y z z0 u v w w0

Note: (2) requires that f be defined at all points in some deleted neighborhood of z0 such a deleted neighborhood always exists when z0 is an interior point of a region on which is defined. We can extend the definition of limit to the case in which z0 is a boundary point of the region by agreeing that left of (2) be satisfied by only those points z that lie in both the region and the domain Example 1. show if

For any such z and any positive number whenever y v z x u

When a limit of a function exists at a point , it is unique. If not, suppose , and Then Let But Hence is a nonnegative constant, and can be chosen arbitrarily small.

Ex 2. If (4) then does not exist. show: since a limit is unique, limit of (4) does not exist. (2) provides a means of testing whether a given point w0 is a limit, it does not directly provide a method for determining that limit.

12. Theorems on limits Thm 1. Suppose that Then iff pf : ” ”

since and “ ” But

Thm 2. suppose that pf: utilize Thm 1. for (9). use Thm 1. and (7)

An immediate consequence of Thm. 1: have the limits An immediate consequence of Thm. 1: by property (9) and math induction. (11) 利用 whenever

13. Limits involving the point at Infinity It is sometime convenient to include with the complex plane the point at infinity, denoted by , and to use limits involving it. Complex plane + infinity = extended complex plane. N P z O complex plane passing thru the equator of a unit sphere. To each point z in the plane there corresponds exactly one point P on the surface of the sphere. intersection of the line z-N with the surface. north pole To each point P on the surface of the sphere, other than the north pole N, there corresponds exactly one point z in the plane.

By letting the point N of the sphere correspond to the point at infinity, we obtain a one-to-one correspondence between the points of the sphere and the points of the extended complex plane. upper sphere exterior of unit circle points on the sphere close to N neighborhood of

Ex 2.

Ex 3.

14. Continuity A function f is continuous at a point z0 if (1) (2) (3) ((3) implies (1)(2)) if continuous at z0, then are also continuous at z0. So is

A composition of continuous function is continuous. A polynomial is continuous in the entire plane because of (11), section 12. p.37 A composition of continuous function is continuous. z w f g If a function f(z) is continuous and non zero at a point z0, then throughout some neighborhood of that point. when let if there is a point z in the at which then , a contradiction.

From Thm 1., sec12. a function f of a complex variable is continuous at a point iff its component functions u and v are continuous there. Ex. The function is continuous everywhere in the complex plane since are continuous (polynomial) cos, sin, cosh, sinh are continuous real and imaginary component are continuous complex function is continuous.

15. Derivatives Let f be a function whose domain of definition contain a neighborhood of a point z0. The derivative of f at z0, written , is provided this limit exists. f is said to be differentiable at z0.

Ex1. Suppose at any point z since a polynomial in . Ex2. when thru on the real axis , Hence if the limit of exists, its value = when thru on the imaginary axis. , limit = if it exists.

since limits are unique, , if is to exist. observe that when exists only at , its value = 0 Example 2 shows that a function can be differentiable at a certain point but nowhere else in any neighborhood of that point. Re are continuous, partially Im differentiable at a point. but may not be differentiable there.

is continuous at each point in the plane since its components are continuous at each point. (前一節) not necessarily continuity derivative exists. existence of derivative continuity.

Differentiation Formulas n a positive integer.

(F continuous at z)

f has a derivative at z0 g has a derivative at f(z0) F(z)=g[f (z)] has a derivative at z0 and chain rule (6) pf of (6) choose a z0 at which f’(z0) exists. let w0 = f (z0) and assume g’(w0) exists. Then, there is of w0 such that we can define a function , with and (7) Hence is continuous at w0

valid even when since exists and therefore f is continuous at z0, then we can have f (z) lies in substitute w by f (z) in (9) when z in (9) becomes since f is continuous at z0 , is continuous at is continuous at z0 , and since so (10) becomes

17. Cauchy-Riemann Equations Suppose that writing Then by Thm. 1 in sec 12 where

Let tend to (0,0) horizontally through . i.e., Let tend to (0,0) vertically thru , i.e. , then (6)= (7) Cauchy-Riemann Equations.

Thm : suppose exists at a point Then Ex 1. Cauchy-Riemann equations are Necessary conditions for the existence of the derivative of a function f at z0. Can be used to locate points at which f does not have a derivative.

Ex 2. does not exist at any nonzero point. The above Thm does not ensure the existence of f ’(z0) (say)

18. Sufficient Conditions For Differentiability but not Thm. Let be defined throughout some neighborhood of a point suppose exist everywhere in the neighborhood and are continuous at . Then, if

Thus where Now in view of the continuity of the first-order partial derivatives of u and v at the point

where and tend to 0 as in the -plane. (3) assuming that the Cauchy-Riemann equations are satisfied at , we can replace in (3), and divide thru by to get also tends to 0, as The last term in(4) tends to 0 as

Ex 1. everywhere, and continuous. exists everywhere, and Ex 2. has a derivative at z=0. can not have derivative at any nonzero point.

19. Polar Coordinates Suppose that exist everywhere in some neighborhood of a given non-zero point z0 and are continuous at that point. also have these properties, and ( by chain rule ) Similarly,

from (2) (5), Thm. p53…

Ex : Consider at any non-zero point exists

20. Analytic Functions A function f of the complex variable z is analytic in an open set if it has a derivative at each point in that set. f is analytic at a point z0 if it is analytic in a neighborhood of z0. Note: is analytic at each non-zero point in the finite plane is not analytic at any point since its derivative exists only at z = 0 and not throughout any neighborhood. - An entire function is a function that is analytic at each point in the entire finite plane. Ex: every polynomial. If a function f fails to be analytic at a point z0 but is analytic at some point in every neighborhood of z0 , then z0 is called a singular point, or singularity, of f .

Ex: z=0 is a singular point of has no singular point since it is no where analytic. Sufficient conditions for analyticity continuity of f throughout D. satisfication of Cauchy-Riemann equation. Sum, Product of analytic functions is/are analytic. Quotient of analytic functions is/are analytic . if denominator A composition of two analytic function is analytic

Thm : If everywhere in a domain D, then must be constant throughout D. pf : write in D at each point in D. are x and y components of vector grad u. grad u is always the zero vector. u is constant along any line segment lying entirely in D.

21. Reflection Principle predicting when the reflection of f (z) in the real axis corresponds to the reflection of z. Thm: f analytic in domain D which contains a segment of the x axis and is symmetric to that axis. Then for each point z in the domain iff f (x) is real for each point x on the segment. pf: “ ” f (x) real on the segment Let write

Now, since f (x+i t) is an analytic function of x + i t. From (4), From (5), Similarly, From (5), since , continuous F(z) is analytic in D.

Since f (x) is real on the segment of the real axis lying in D, This is at each point z = x on the segment. (6) From Chap 6 (sec.58) A function f that is analytic in D is uniquely determined by its value along any line segment lying in D. (6) holds throughout D. “ ”

expand (7): for (x,0) on real axis is real on the segment of real axis lying in D. Ex: for all z. since are real when x is real. do not have reflection property since are not real when x is real.

22. Haronic Functions A real-valued function H of two real variables x and y is said to be harmonic in a given domain of the xy plane if, throughput that domain, it has continuous partial derivatives of the first and second order and satisfies the partial differential equation known as Laplace's equation. Applications: Temperatures T(x,y) in thin plate. Electrostatic potential in the interior of a region of 3-D space. Ex 1. is harmonic in any domain of the xy plane. Thm 1. If a function is analytic in a domain D, then its component functions u and v are harmonic in D.

Pf: Assume f analytic in D Pf: Assume f analytic in D. (then its real and imaginary components have continuous partial derivatives of all orders in D) “proved in chap 4” the continuity of partial derivatives of u and v ensures this (theorem in calculus) Ex 2. is entire. in Ex 1. must be harmonic. Hence

Ex 3. is entire. so is f (z)g(z). ( f (z) in Ex 2 ) is harmonic. If u , v are harmonic in a domain D, and their first-order partial derivatives satisfy Cauchy-Riemann equation throughout D, v is said to be harmonic conjugate of u. (u is not necessary harmonic conjugate of v ) Thm 2 v is a harmonic conjugate of u. Ex 4: and are real & imag. parts of is a harmonic conjugate of . But is not analytic.

If v is a harmonic conjugate of u in a domain D, then -u is a harmonic conjugate of v in D. It can be shown that [Ex 11(b).] if two functions u and v are to be harmonic conjugates of each other, then both u and v must be constant functions. In chap 9, shall show that a function u which is harmonic in a domain of certain type always has a harmonic conjugate. Thus, in such domains, every harmonic function is the real part of an analytic function. A harmonic conjugate, when it exists, is unique except for an additive constant.

Ex 5. To obtain a harmonic conjugate of a given harmonic function