9 Multiple valued functions

9.1 Domains of univalence

Let's review the inverse function theorem in advanced calculus:

Given a function F : S ! Rⁿ; S is an open set and S Rⁿ: F 2 C¹(S) and F = (f₁; f₂; :::; f_n) : a vector-valued function.

For a point a 2 S; b = F (a) ; ^{in case} J_F (a) = @ (f₁; f₂; :::; f_n)

@ (x₁; x₂; :::; x_n) (a) 6= 0

: The Jacobian of F at a, then we must have open sets A and B such that a 2 A ^and b 2 B; F : A ! B is de ned and there is an inverse G : B ! A ^of F; G 2 C¹(B) :

Def. f (z) is ”univalent” on a domain G if f (z) is 1-1 and analytic on G; or G is called a ”domain of univalence” for f (z) : Theorem 66 f (z) is univalent on a domain G

) f⁰(z) 6= 0; 8z 2 G:

pf. The result is immediate followed by the inverse function theorem.

Theorem 67 ! = f (z) = u (x; y) + iv (x; y) is univalent on a domain G; and E is the image of G under f (z) :

) E is also a domain.

Complex Analysis 115 pf. i). To claim that E is connected.

Consider any two points !₁ and !₂ of the set E; there exist z₁ and z₂ on G such that

!_j = f (z_j) ; j = 1; 2:

Since G is connected, there is a curve C : z = z (t) ; t 2 [a; b]

such that

z (a) = z₁; z (b) = z₂ and C G:

Now consider the curve : ! = ! (t) = f (z (t)) ; t 2 [a; b] : Because f is univalent, it is clear that E; and that

! (a) = !₁; ! (b) = !₂: Therefore E is connected. ]

ii). To show that E is open.

For any !₀ = u₀+ iv₀ 2 E; there exists z₀ = x₀+ iy₀ 2 G such that f (z₀) = !₀: Apply the inverse function theorem to the function

F (x; y) = (U; V ) (x; y)

:vector-valued function and F 2 C¹(G) : Note that we convert f into F; and the analyticity part of f becomes F 2 C¹(G) : Then we have

J_F (x₀; y₀) =

@U

@x; ^@U_@y

@V

@x; ^@V_@y

(x0;y0)

= U_x; V_x V_x; U_{x (x}

0;y₀)

;

Complex Analysis 116 and hence

J_F (x₀; y₀) = U_x²(x₀; y₀) + V_x²(x₀; y₀) = jf⁰ (z₀)j² > 0:

Again, by inverse function theorem, there exist two open sets A and B such that (x₀; y₀) 2 A ^and (u₀; v₀) 2 B; ^and

F : A ^{1 1}!

onto B;

i. e. f (A) = B E: Since for arbitrary !₀ = f (z₀) 2 B E;

we must have E being an open set. ] Analogy: In the proof above, we have

f : G ^{1 1}!

onto E:

Then it is natural to set ' as the inverse of f; and ' : E ^{1 1}!

onto G:

Similarly as in the above, for any !₀ 2 E; we may also set a vector function L : B ^{1 1}!

onto A being the inverse of F such that L (u; v) = (X; Y ) (u; v) 2 C¹(B) : •

Theorem 68 Assume that ! = f (z) ; the domain G and domain E are the same as in the previous theorem. If z = ' (!) is the inverse of ! = f (z) ; we nd that ' (!) is univalent on E and

'⁰ (!) = 1 f⁰ (z):

Complex Analysis 117

Ex. Consider the function ! = zⁿ for its univalent domain and corresponding range. on the z plane. To see this problem on the other hand, consider z = reⁱ : Then ! = rⁿeⁱⁿ ; i. e.

= rⁿ; = n + 2k ; k 2 Z:

Now let's choose G = reⁱ j 0 < < ²_n ; r > 0 : a wedge on the z plane. Then it is easy to see that

! : G ! E

is a 1-1 mapping, and ! is univalent on G; where E is the

Complex Analysis 118 complex ! plane cut along the non-negative real axis.

Figure.

Similarly, every wedge reⁱ j < < + ²_n ; r > 0 and 2 Rg is also a univalent domain for zⁿ:

Figure.

Hence we nd that zⁿ is univalent on the domain reⁱ j < < + ²_n ; r > 0 and 2 R ; and the corre-sponding inverse is

z = pⁿ

eⁱⁿ;

where n < < n + 2 : We may denote this z by pⁿ

! and nd that

dpⁿ

!

d! = 1

nz^{n 1} = z nzⁿ =

pn

! n! :

Ex. Consider the function ! = e^z for its univalent domain and corresponding range.

Sol. Consider z = x + iy; x; y 2 R: Then the mapping can

Complex Analysis 119 be interpreted as

! = e^x+iy = e^x (cos y + i sin y) ;

where j!j = e^{x and} arg ! = y: Consider z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂: Then let us analyze the mapping as follows:

i). if x₁ 6= x²; e^z1 6= e^z2:

ii). if x₁ = x₂ = x; look at the difference:

!₂ !₁ = e^x e^iy2 e^iy1

= e^x [(cos y₂ cos y₁) + i (siny₂ sin y₁)]

= e^x 2 sin y₁ + y₂

2 sin y₂ y₁ 2 +2i cos y₁ + y₂

2 sin y₂ y₁ 2

= 2ie^x sin y₂ y₁ 2

h

cos y₁ + y₂

2 + i sin y₁ + y₂ 2

i

= 2ie^x sin y₂ y₁

2 exph

i y₁ + y₂ 2

i :

Thus we nd that !₂ = !₁ iff. sin ^y2₂^y1 = 0; i. e. y₂ = y₁+2k ; k 2 Z:

Figure.

Complex Analysis 120 Let's choose G = fzj 0 < Im z < 2 g ; ^then

! : G ! E

is univalent on G, and E is the ! plane cut along the non-negative real axis. To generalize it, set

G = fzj < Im z < + 2 ; 2 Rg : Then the function ! : G ! E is univalent on G:

If consider the inverse function z of ! : ln ! = z = x + iy;

where

ln j!j = x; arg ! = y:

i. e.

ln ! = ln j!j + i arg !; < arg ! < + 2 : Def. For ! 6= 0; ^{we de ne}

ln ! = ln j!j + i arg !;

which is a multi-valued function.

Ex. Evaluate ln i:

Sol. By de nition,

ln i = ln jij + i arg i = 0 + i

2 + 2k ; where k 2 Z:

Ex. Compute ln 1 + ip 3 :

Complex Analysis 121 Sol. Again, by using de nition, we nd

ln 1 + ip

3 = ln 1 + ip

3 + i arg 1 + ip 3

= ln 2 + i 2

3 + 2k ; k 2 Z:

Corollary 69

d ln !

d! = 1

exp z = 1

!:

Complex Analysis 122 9.2 Branches, branch cuts and branch points

9.2.1 Branches, etc.

Def. A branch of a multiple-valued function F is any single-valued function f which is analytic on some domain at each z of which the value f (z) is one of the values F (z) s.

Ex. Find the branches of the inverse function z = pⁿ

! of

! = zⁿ:

Sol. Consider the following domains:

G₀ : 0 < < ²_n ;

All of these are non-overlapping

domains of univalence of the function zⁿ: So we consider the function de ned as

zⁿ : G_k ! E;

where k = 0; 1; 2; :::; n 1 and E is the ! plane cut along the nonnegative real axis.

Denote the inverse(s) of the above function by pn

! _k : E ! G^k; k = 0; 1; 2; :::; n 1:

The n single-valued functions fpⁿ

!g0 ; fpⁿ

!g1 ; ...., fpⁿ

!gn 1

are called the ”branches” of the multiple-valued function z = pⁿ

!: \

Complex Analysis 123 Ex. Find the branches of the inverse function z = ln ! of

! = exp z:

Sol. Again, consider the domains:

G₀ : 0 < Im z < 2 ;

There are in nitely

many non-overlapping domains of univalence for the function

! = exp z; k 2 Z: So the function

exp z : G_k ! E

is univalent on G_k; where E is the ! plane cut along the non-negative real axis.

Denote the inverse(s) of the above function by fln !g_k : E ! G^k ; k 2 Z:

Then each fln !gk is a single-valued function and is called a

”branch” of the multiple-valued function z = ln !: \

Def. When certain region of the domain is avoided during the mapping of a branch function, we call the speci c avoided region the ”branch cut.”

Remark 16 It should be kept in mind that the notion of a branch

Complex Analysis 124 of a multiple-valued function is intimately related to the choice of the corresponding domain of univalence, and hence inevitably contains an element of arbitrariness.

e. g. If in the example ! = zⁿ; the domain of univalence G_k is chosen instead as

G⁰_k : 2k 1

n < < 2k + 1 n

, k = 0; 1; 2; :::; n 1; then corresponding range E⁰ becomes the

! plane cutting along the non-positive real axis (or the branch cut becomes the non-positive real axis instead of the non-negative real axis.)

9.2.2 Branch points

Def. A point with the property that a circuit around any Jordan curve with on its interior carries every branch of of a given multiple-valued function into another branch is called a branch point.

Def. If a nite number (say, n 2 N) of circuit(s) aroud a branch point in the same direction carries every branch of the function into itself, then is said to be ”of nite order n:”

Ex. Let's consider a simple Jordan curve in the mapping of ! = zⁿ: Move ! along with initial and nal point

Complex Analysis 125 Note that the same point z₀ can also be expressed as

z₀ = pⁿ

The curve lies totally inside one branch— no matter what the traveling direction of z is.

ii). ! = 0 lies inside of :

If z travels in the counterclockwise direction, after crossing arg ! = 0; the ”1st nal point” becomes

z₁ = pⁿ

0 cos ⁰ + 2

n + i sin ⁰ + 2

n :

Note that the corresponding values of pⁿ

! changes continuously from branch fpⁿ

!g0 to fpⁿ

!g1: On the other hand, when z travels in the clockwise direction, the 1st nal point becomes

z_{n 1} = pⁿ

0 cos ⁰ 2

n + i sin ⁰ 2

n ;

and the corresponding values of pⁿ

! changes continuously from branch fpⁿ

!g0 to fpⁿ

!gn 1 : We may also see that in this case,

Complex Analysis 126 the order of branch point ! = 0 is n:

iii). ! = 0 lies outside of and cross arg ! = 0:

The curve moves onto other branches of fpⁿ

!gk before eventually returning to z₀: This case is more complicated.

Figure.

Ex. Turn our attention to ! = exp z; and again consider a Jordan curve : The ”initial- nal” point z₀ is denoted by

z₀ = ln j!⁰j + i 0;

where !₀ = exp z₀ = j!⁰j exp (i 0) 2 E; ^and arg !₀ = ₀: The corresponding branch of the logarithmic function is fln !g_k :

Similarly as in the above example, let's discuss 3 different relations between and the point ! = 0. We nd that in case ii). and iii)., no matter how many circuits we make around in a given direction, the branch fln !g_k is never carried back to the original itself— instead it keeps generating new branches of ln !:

We then call the branch point ! = 0 logarithmic or of in nite order.

Complex Analysis 127 Figure. ! = 0 outside :

Figure. ! = 0 intside :

Discussion: In the above 2 examples, consider the point

! = 1 in the sense of Riemann sphere. Check to see if it is a branch point as well as check its order in either cases.

Def. (Complement from texbook) If a function f is analytic at a point z 2 D = fzj 0 < jz aj < "g : a puctured disk. f does not extend analytically on D but has an analytic continuation along any path on D; we say that f has a ”branch point” at z = a:

Remark 17 We can see that every point of a branch cut is not necessarily a branch point for a speci c multiple-valued function.

Complex Analysis 128 9.3 Riemann surfaces

Def. A Riemann surface is a set R with a collection of subsets fU g and complex-valued functions z (p) ; p 2 U ; ^{such that}

(1.1) z (p) : U ^{1 1}! z (U ) on the complex plane, 8 ; (1.2) z z ¹ : z (U \ U ) ! z (U \ U ) is analytic

whenever it is de ned;

(1.3) R is connected, i. e. 8p; q 2 R; 9 ¹; ₂; :::; _m such that

p 2 U 1; q 2 U m ^ U j \ U j+1 6= ; where j = 1; 2; 3; :::; m 1;

(1.4) R is a “Hausdorff space,” i. e. 8p; q 2 R; p 6= q;

p 2 U ^and q 2 U ; there exist small disks D₀ and D₁ centered at z (p) and z (q) ; respectively, such that

z ¹(D₀) ; z ¹(D₁) 2 R ^ z ¹ (D₀) \ z ¹(D₁) = : Note: We refer to U as a coordinate patch and to z (p) as the coordinate map on U ; mapping U onto z (U ) : The composition z z ¹ corresponds to a change of variable. If D₀ is a disk on z (U ) ; we refer !₀ = z ¹(D₀) as a coordinate disk on R: Also note that

z : z ¹(D₀) ^{1 1}!

onto D₀:

Complex Analysis 129 Figure.

Ex. Consider the Riemann surface in the case of z = pⁿ

!:

Sol. Figure.

Let's consider a n sheet structure:

Figure.

Paste ₀ to ⁺₁ ; ₁ to ⁺₂ ; ::::; _{n 2} to ⁺_{n 1}; and _{n 1} to

+

0 : The ”n sheeted structure” is the Riemann surface S of the multiple-valued function z = pⁿ

!: To de ne z = pⁿ

! on S; set pn

! E_k = pⁿ

! k ; and pⁿ

! : S ^{1 1}!

onto z plane.

Complex Analysis 130 Note that pⁿ

! is an analytic function except at the origin, since dpⁿ

! d! =

pn

! n! :

Ex. Consider the Riemann surface in the case of z = ln !:

Sol. Figure.

Similarly to the above case, there exists in nitely many sheets E_k; k 2 Z: The lower edge of the cut E_k is pasted to the upper edge of the cut on the sheet E_k+1: Then we obtain a Riemann surface for ln !:

To de ne ln ! on S; simply set

ln !j_Ek = fln !g_k ^and ln ! : Snf0g ^{1 1}!

onto z plane.

Note that the function ln ! is analytic on domain Snf0g; ^since d ln !

d! = 1

exp z = 1

!:

Complex Analysis 131

在文檔中 Complex Analysis Lecturenotes Nai-Sher Yeh (頁 114-131)