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Contour integrals and Cauchy’s Theorem

3.1

Line integrals of complex functions

Our goal here will be to discuss integration of complex functions f (z) = u + iv, with particular regard to analytic functions. Of course, one way to think of integration is as antidiﬀerentiation. But there is also the deﬁnite integral. For a function f (x) of a real variable x, we have the integral b f (x) dx. In case f (x) = u(x) + iv(x) is a complex-valued function of a a real variable x, the deﬁnite integral is the complex number obtained by b integrating the real and imaginary parts of f (x) separately, i.e. b b

u(x) dx + i a f (x) dx = a v(x) dx. For vector ﬁelds F = (P, Q) in the plane we have a the line integral

P dx+Q dy, where C is an oriented curve. In case P and

C

Q are complex-valued, in which case we call P dx + Q dy a complex 1-form, we again deﬁne the line integral by integrating the real and imaginary parts separately. Next we recall the basics of line integrals in the plane:

1. The vector ﬁeld F = (P, Q) is a gradient vector ﬁeld g, which we can write in terms of 1-forms as P dx + Q dy = dg, if and only if

C P dx+Q dy only depends on the endpoints of C, equivalently if and only if C P dx+Q dy = 0 for every closed curve C. If P dx+Q dy = dg, and C has endpoints z0 and z1 , then we have the formula dg = g(z1 ) − g(z0 ).

P dx + Q dy =

C

C

2. If D is a plane region with oriented boundary ∂D = C, then

P dx + Q dy =

C

D

∂Q ∂P

−

∂x

∂y

dxdy.

3. If D is a simply connected plane region, then F = (P, Q) is a gradient vector ﬁeld g if and only if F satisﬁes the mixed partials condition

∂Q

∂P

=

.

∂x

∂y

(Recall that a region D is simply connected if every simple closed curve in

D is the boundary of a region contained in D. Thus a disk {z ∈ C : |z| < 1}

1

is simply connected, whereas a “ring” such as {z ∈ C : 1 < |z| < 2} is not.)

In case P dx + Q dy is a complex 1-form, all of the above still makes sense, and in particular Green’s theorem is still true.

We will be interested in the following integrals. Let dz = dx + idy, a complex 1-form (with P = 1 and Q = i), and let f (z) = u + iv. The expression f (z) dz = (u + iv)(dx + idy) = (u + iv) dx + (iu − v) dy

= (udx − vdy) + i(vdx + udy) is also a complex 1-form, of a very special type.

Then we can deﬁne

f (z) dz for any reasonable closed oriented curve C. If C is a parametrized

C

curve given by r(t), a ≤ t ≤ b, then we can view r (t) as a complex-valued curve, and then b f (r(t)) · r (t) dt,

f (z) dz = a C

where the indicated multiplication is multiplication of complex numbers

(and not the dot product). Another notation which is frequently used is the following. We denote a parametrized curve in the complex plane by z(t), a ≤ t ≤ b, and its derivative by z (t). Then b f (z) dz =

C

f (z(t))z (t) dt. a For example, let C be the curve parametrized by r(t) = t + 2t2 i, 0 ≤ t ≤ 1, and let f (z) = z 2 . Then

1

z 2 dz =

C

1

(t + 2t2 i)2 (1 + 4ti) dt =

0

(t2 − 4t4 + 4t3 i)(1 + 4ti) dt

0

1

[(t2 − 4t4 − 16t4 ) + i(4t3 + 4t3 − 16t5 )] dt

=

0

3

= t /3 − 4t5 + i(2t4 − 8t6 /3)]1 = −11/3 + (−2/3)i.

0

For another example, let let C be the unit circle, which can be eﬃciently parametrized as r(t) = eit = cos t + i sin t, 0 ≤ t ≤ 2π, and let f (z) = z .

¯

Then r (t) = − sin t + i cos t = i(cos t + i sin t) = ieit . d Note that this is what we would get by the usual calculation of eit . Then dt 2π

C

2π

eit · ieit dt =

z dz =

¯

0

0

2π

e−it · ieit dt =

i dt = 2πi.

0

2

One ﬁnal point in this section: let f (z) = u + iv be any complex valued function. Then we can compute f , or equivalently df . This computation is important, among other reasons, because of the chain rule: if r(t) =

(x(t), y(t)) is a parametrized curve in the plane, then d f (r(t)) = dt f · r (t) =

∂f dx ∂f dy

+

.

∂x dt

∂y dt

d

(Here · means the dot product.) We can think of obtaining f (r(t)) roughly dt ∂f

∂f

by taking the formal deﬁnition df = dx + dy and dividing both sides

∂x

∂y by dt.

Of course we expect that df should have a particularly nice form if f (z) is analytic. In fact, for a general function f (z) = u + iv, we have df =

∂u

∂v

+i

∂x

∂x

dx +

∂u

∂v

+i

∂y

∂y

dy

and thus, if f (z) is analytic, df =

=

∂u

∂v

+i

∂x

∂x

∂u

∂v

+i

∂x

∂x

∂v

∂u

+i dy ∂x

∂x

∂u

∂v

+i idy =

∂x

∂x

dx + − dx +

∂u

∂v

+i

∂x

∂x

(dx + idy) = f (z) dz.

Hence: if f (z) is analytic, then df = f (z) dz and thus, if z(t) = (x(t), y(t)) is a parametrized curve, then d f (z(t)) = f (z(t))z (t) dt This is sometimes called the chain rule for analytic functions. For example, if α = a + bi is a complex number, then applying the chain rule to the analytic function f (z) = ez and z(t) = αt = at + (bt)i, we see that d αt e = αeαt . dt 3

3.2

Cauchy’s theorem

Suppose now that C is a simple closed curve which is the boundary ∂D of a f (z) dz.

region in C. We want to apply Green’s theorem to the integral

C

Working this out, since

f (z) dz = (u + iv)(dx + idy) = (u dx − v dy) + i(v dx + u dy), we see that

−

f (z) dz =

D

C

∂v

∂u

−

∂x ∂y

dA + i

D

∂u ∂v

−

∂x ∂y

dA.

Thus, the integrand is always zero if and only if the following equations hold:

∂v

∂u

=− ;

∂x

∂y

∂u

∂v

=

.

∂x

∂y

Of course, these are just the Cauchy-Riemann equations! This gives:

Theorem (Cauchy’s integral theorem): Let C be a simple closed curve which is the boundary ∂D of a region in C. Let f (z) be analytic in D. Then f (z) dz = 0.

C

Actually, there is a stronger result, which we shall prove in the next section: Theorem (Cauchy’s integral theorem 2): Let D be a simply connected region in C and let C be a closed curve (not necessarily simple) contained in D. Let f (z) be analytic in D. Then f (z) dz = 0.

C

Example: let D = C and let f (z) be the function z 2 + z + 1. Let C be the unit circle. Then as before we use the parametrization of the unit circle given by r(t) = eit , 0 ≤ t ≤ 2π, and r (t) = ieit . Thus

2π

C

2π

(e2it + eit + 1)ieit dt = i

f (z) dz =

0

(e3it + e2it + eit ) dt.

0

4

It is easy to check directly that this integral is 0, for example because terms

2π

such as 0 cos 3t dt (or the same integral with cos 3t replaced by sin 3t or cos 2t, etc.) are all zero.

On the other hand, again with C the unit circle,

C

1 dz = z 2π

2π

e−it ieit dt = i

0

dt = 2πi = 0.

0

The diﬀerence is that 1/z is analytic in the region C−{0} = {z ∈ C : z = 0}, but this region is not simply connected. (Why not?) f (z) dz = 0

Actually, the converse to Cauchy’s theorem is also true: if

C

for every closed curve in a region D (simply connected or not), then f (z) is analytic in D. We will see this later.

3.3

Antiderivatives

If D is a simply connected region, C is a curve contained in D, P , Q are de∂Q

∂P

ﬁned in D and

=

, then the line integral

P dx+Q dy only depends

∂x

∂y

C

on the endpoints of C. However, if P dx + Q dy = dF , then

P dx + Q dy

C

only depends on the endpoints of C whether or not D is simply connected.

We see what this condition means in terms of complex function theory: Let f (z) = u + iv and suppose that f (z) dz = dF , where we write F in terms of its real and imaginary parts as F = U + iV . This says that

(u dx − v dy) + i(v dx + u dy) =

∂U

∂U

dx + dy + i

∂x

∂y

∂V

∂V

dx + dy .

∂x

∂y

Equating terms, this says that

∂V

∂U

=

∂x

∂y

∂U

∂V

v=−

=

.

∂y

∂x

u=

In particular, we see that F satisﬁes the Cauchy-Riemann equations, and its complex derivative is

F (z) =

∂U

∂V

+i

= u + iv = f (z).

∂x

∂x

5

We say that F (z) is a complex antiderivative for f (z), i.e. F (z) = f (z). In this case

∂u

∂2U

∂2V

∂v

=

=

=

;

2

∂x

∂x

∂x∂y

∂y

∂u

∂2V

∂2U

∂v

=−

=

=− .

2

∂y

∂y

∂x∂y

∂x

It follows that, if f (z) has a complex antiderivative, then f (z) satisﬁes the

Cauchy-Riemann equations: f (z) is necessarily analytic.

Thus we see:

Theorem: If the 1-form f (z) dz is of the form dF , or equivalently the vector ﬁeld (u + iv, −v + iu) is a gradient vector ﬁeld (U + iV ), then both

F (z) and f (z) are analytic, and F (z) is a complex antiderivative for f (z):

F (z) = f (z). Conversely, if F (z) is a complex antiderivative for f (z), then

F (z) and f (z) are analytic and f (z) dz = dF .

The theorem tells us a little more: Suppose that F (z) is a complex antiderivative for f (z), i.e. F (z) = f (z). If C has endpoints z0 and z1 , and is oriented so that z0 is the starting point and z1 the endpoint, then we have the formula f (z) dz = dF = F (z1 ) − F (z0 ).

C

C

For example, we have seen that, if C is the curve parametrized by r(t) = z 2 dz = −11/3+(−2/3)i. But z 3 /3

t+2t2 i, 0 ≤ t ≤ 1 and f (z) = z 2 , then

C

is clearly an antiderivative for z 2 , and C has starting point 0 and endpoint

1 + 2i. Hence z 2 dz = (1 + 2i)3 /3 − 0 = (1 + 6i − 12 − 8i)/3 = (−11 − 2i)/3,

C

which agrees with the previous calculation.

When does an analytic function have a complex antiderivative? From vector calculus, we know that f (z) dz = dF if and only if C f (z) dz only depends on the endpoints of C, if and only if C f (z) dz = 0 for every closed curve C. In particular, if C f (z) dz = 0 for every closed curve C then f (z) is analytic (converse to Cauchy’s theorem).

If f (z) is analytic in a simply connected region D, then the fact that f (z) dz = P dx + Q dy satisﬁes ∂Q/∂x = ∂P/∂y (here P and Q are complex valued) says that (P, Q) is a gradient vector ﬁeld, or equivalently that f (z) dz = dF , in other words that f (z) has an antiderivative. Hence:

6

Theorem: Let D be a simply connected region and let f (z) be an analytic function in D. Then there exists a complex antiderivative F (z) for f (z).

Fixing a base point p0 ∈ D, a complex antiderivative F (z) for f (z) is given f (z) dz, where f (z) is any curve in D joining p0 to z.

by

C

As a consequence, we see that, if D is simply connected, f (z) is analytic f (z) dz = 0 (Cauchy’s integral

in D and C is a closed curve in D, then

C

theorem 2), since f (z) dz = dF , where F is a complex antiderivative for f (z), and hence f (z) dz =

C

dF = 0,

C

by the Fundamental Theorem for line integrals.

1

dz = 2πi = 0, where C z C is the unit circle. The antiderivative of 1/z is log z, and so the expected answer (viewing the unit circle as starting at 1 = e0 and ending at e2πi = 1 is log 1 − log 1. But log is not a single-valued function, and in fact as z = eit turns along the unit circle, the value of log changes by 2πi. So the correct answer is really log 1 − log 1, viewed as log e2πi − log e0 = 2πi − 0 = 2πi.

Of course, 1/z is analytic except at the origin, but {z ∈ C : z = 0} is not simply connected, and so 1/z need not have an antiderivative.

The real point, however, in the above example is something special about log z, or 1/z, but not the fact that 1/z fails to be deﬁned at the origin. We could have looked at other negative powers of z, say z n where n is a negative integer less than −1, or in fact any integer = −1. In this case, z n has an antiderivative z n+1 /(n + 1), and so by the fundamental theorem for line

From this point of view, we can see why

z n dz = 0 for every closed curve C. To see this directly for the

integrals

C

case n = −2 and the unit circle C,

2π

z −2 dz =

C

2π

e−2it ieit dt = i

0

e−it dt = 0.

0

This calculation can be done somewhat diﬀerently as follows. Let r(t) = eαt , where α is a nonzero complex number. Then, by the chain rule for analytic functions, an antiderivative for the complex curve r(t) is checked to be s(t) =

eαt dt =

7

1 αt e . α Hence, b eαt dt = a 1 αb e − eαa . α z n dz = 0 for every integer n = −1, where

In general, we have seen that

C

C is a closed curve. To verify this for the case of the unit circle, we have

2π

2π

0

0

C

e(n+1)it dt =

enit ieit dt = i

z n dz =

i e(n+1)it i(n + 1)

2π

0

i

1

= e2(n+1)πi − e0 =

(1 − 1) = 0. i(n + 1) n+1 Finally, returning to 1/z, a calculation shows that

1

dz = z x dx y dy

+

x2 + y 2 x2 + y 2

+i

−y dx x dy

+

x2 + y 2 x2 + y 2

.

The real part is the gradient of the function 1 ln(x2 + y 2 ) = d ln r. But the

2

imaginary part corresponds to the vector ﬁeld

F=

−y x , x2 + y 2 x2 + y 2

,

which is a standard example of a vector ﬁeld F for which Green’s theorem fails, because F is undeﬁned at the origin. In fact, in terms of 1-forms, x dy

−y dx

+ 2

= d arg z = dθ.

2 + y2 x x + y2

In the next section, we will see how to systematically use the fact that the integral of 1/z dz around a closed curve enclosing the origin to get a formula for the value of an analytic function in terms of an integral.

3.4

Cauchy’s integral formula

Let C be a simple closed curve in C. Then C = ∂R for some region R (in other words, C is simply connected). If z0 is a point which does not lie on

C, we say that C encloses z0 if z0 ∈ R, and that C does not enclose z0 if z0 ∈ R. For example, if C is the unit circle, then C is the boundary of the

/

unit disk B = {z : |z| < 1}. Thus C encloses a point z0 if z0 lies inside the unit disk (|z0 | < 1), and C does not enclose z0 if z0 lies outside the unit disk

(|z0 | > 1). We always orient C by viewing it as ∂R and using the orientation coming from the statement of Green’s theorem.

8

Theorem (Cauchy’s integral formula): Let D be a simply connected region in C and let C be a simple closed curve contained in D. Let f (z) be analytic in D. Suppose that z0 is a point enclosed by C. Then f (z0 ) =

1

2πi

C

f (z) dz. z − z0

For example, if C is a circle of radius 5 about 0, then

2

C

ez dz = 2πie4 . z−2 2

But if C is instead the unit circle, then

C

ez dz = 0, as follows from z−2 Cauchy’s integral theorem.

Before we discuss the proof of Cauchy’s integral formula, let us look at the special case where f (z) is the constant function 1, C is the unit circle, and z0 = 0. The theorem says in this case that

1

2πi

1 = f (0) =

C

1 dz, z

as we have seen. In fact, the theorem is true for a circle of any radius: if

Cr is a circle of radius r centered at 0, then Cr can be parametrized by reit ,

0 ≤ t ≤ 2π. Then

Cr

1 dz = z 2π

0

1 ireit dt = i reit independent of r. The fact that

Cr

2π

dt = 2πi,

0

1 dz is independent of r also follows z from Green’s theorem.

The general case is obtained from this special case as follows. Let C =

∂R, with R ⊆ D since D is simply connected. We know that C encloses z0 , which says that z0 ∈ R. Let Cr be a circle of radius r with center z0 . If r is small enough, Cr will be contained in R, as will the ball Br of radius r with center z0 . Let Rr be the region obtained by deleting Br from R.

Then ∂Rr = C − Cr , where this is to be understood as saying that the boundary of Rr has two pieces: one is C with the usual orientation coming from the fact that C is the boundary of R, and the other is Cr with the clockwise orientation, which we record by putting a minus sign in front

9

of Cr . Now z0 does not lie in Rr , so we can apply Green’s theorem to the function f (z)/(z − z0 ) which is analytic in D except at z0 and hence in Rr :

∂Rr

f (z) dz = 0. z − z0

But we have seen that ∂Rr = C − Cr , so this says that

C

f (z) dz − z − z0

Cr

f (z) dz = 0, z − z0

or in other words that

C

f (z) dz = z − z0

Cr

f (z) dz. z − z0

Now suppose that r is small, so that f (z) is approximately equal to f (z0 ) f (z) on Cr . Then the second integral dz is approximately equal to z − z0

Cr

Cr

f (z0 ) dz = f (z0 ) z − z0

Cr

1 dz, z − z0

where Cr is a circle of radius r centered at z0 . Thus we can parametrize Cr by z0 + reit , 0 ≤ t ≤ 2π, and

Cr

1 dz = z − z0

2π

0

as before. Thus f (z0 )

Cr

1 ireit dt = i reit 2π

dt = 2πi,

0

1 dz = 2πif (z0 ), z − z0

f (z) dz is approximately equal to 2πif (z0 ). In

Cr z − z0

C

f (z) fact, this becomes an equality in the limit as r → 0. But dz is

C z − z0 independent of r, and so in fact and so

f (z) dz = z − z0

C

f (z) dz = 2πif (z0 ). z − z0

Dividing through by 2πi gives Cauchy’s formula.

The main theoretical application of Cauchy’s theorem is to think of the point z0 as a variable point inside of the region R such that C = ∂R; note

10

that the z in the formula is a dummy variable. Thus we could equally well write: 1 f (w) f (z) = dw, 2πi C w − z for all z enclosed by C. This description of the analytic function f (z) by an integral depending only on its values on the boundary curve of R turns out to have many very surprising consequences. For example, it turns out that an analytic function actually has derivatives of all orders, not just ﬁrst derivatives, which is very unlike the situation for functions of a real variable.

In fact, every analytic function can be expressed as a power series. This fact can be seen by rewriting Cauchy’s formula above as f (z) =

1

2πi

C

f (w) dw, w(1 − z/w)

1 as a geometric series. The fact that every

1 − z/w analytic function is given by a convergent power series is yet another way of characterizing analytic functions. and then expanding

3.5

Homework

1. Let f (z) = x2 + iy 2 . Evaluate

f (z) dz, where (a) C is the straight

C

line joining 1 to 2 + i; (b) C is the curve (1 + t) + t2 i, 0 ≤ t ≤ 1. Are the results the same? Why or why not might you expect this?

2. Let α = c + di be a complex number. Verify directly that d αt e = αeαt . dt 3. Let D be a region in C and let u(x, y) be a real-valued function on D.

We seek another real-valued function v(x, y) such that f (z) = u + iv is analytic, i.e. satisﬁes the Cauchy-Riemann equations. Equivalently, we want to ﬁnd a function v such that

∂u

∂v

∂u

∂v

=−

and

=

,

∂x

∂y

∂y

∂x

∂u ∂u

,

. Show that

∂y ∂x

F satisﬁes the mixed partials condition exactly when u is harmonic.

Conclude that, if D is simply connected, then F is a gradient vector ﬁeld v and hence that u is the real part of an analytic function. which says that

v is the vector ﬁeld F =

11

−

4. Let C be a circle centered at 4+i of radius 1. Without any calculation,

1

explain why dz = 0.

C z

5. Let C be the curve deﬁned parametrically as follows: z(t) = t(1 − t)et + [cos(2πt3 )]i,

0 ≤ t ≤ 1.

2

ez dz. Be sure to explain your reasoning!

Evaluate the integral

C

6. Let D be a simply connected region in C and let C be a simple closed curve contained in D. Let f (z) be analytic in D. Suppose that z0 is a

1

f (z) point which is not enclosed by C. What is dz? 2πi C z − z0 ez dz, where C is a circle of

C z+1 radius 4 centered at the origin (and oriented counterclockwise).

7. Use Cauchy’s formula to evaluate

8. Let C be the unit circle centered at 0 in the complex plane C and oriented counterclockwise. Evaluate each of the following integrals, and be sure that you can justify your answer by a calculation or a clear and concise explanation.

2

z 4 dz;

(a)

(b)

C

(d)

C

z 2 − 1/3 dz; z+5

(e)

e−z dz; (c) z −5 dz; z − i/2

C

C

1

e−2z dz; (f) dz. 2

C (12z − 5)

C 3z + 2

9. Let D be a simply connected region in C and let C be a simple closed curve contained in D. Let f (z) be analytic in D. Suppose that z0 is a point enclosed by C.

(a) By the usual formulas, show that d dz

f (z) z − z0

=

f (z) f (z)

−

. z − z0 (z − z0 )2

(b) By using the fact that the line integral of a complex function with an antiderivative is zero and the above, conclude that

C

f (z) dz = z − z0

12

C

f (z) dz. (z − z0 )2

(c) Now apply Cauchy’s formula to conclude that f (z0 ) =

1

2πi

13

C

f (z) dz. (z − z0 )2…...

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...Feminism can be roughly defined as a movement that seeks to enhance the quality of women’s lives by impacting the norms and moves of a society based on male dominance and subsequent female subordination. The means of change in the work place, politically, and domestically. Women have come a long way since the 19th century. Women have been trying to prove to the male dominant world that they are equal. They can perform and complete any tasks equal, or in some cases better than man. Feminism has changed the definition of men in many ways. Women in the work place have transposed dramatically since the 19th and mid 20th century. Even if women had any education in the 19th century they were not allow to manifest any of it. It just was not proper for women to give any signs of intelligence and a brain of their own. They were to prepare themselves to become wives and mothers, which were the extent of their entire lives. In the early and mid 20th century some women were starting to be brave and take a stand for themselves. The beginnings of feminism were starting to take its massive role in society. More and more women were getting educated and looking for employment opportunities that had power. Men no longer can be in control of everything. Men in the work place started to feel impotent. But women fed off each other and gave each other strength. They were not looking for just the secretarial jobs; they were taking some men’s jobs and being good at it. They were becoming police......

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...Either you can do it this way or that way “Learn from your own mistakes, or find a master to guide you along”. Everyone in this world is granted only one life and the decisions we make have a great impact on our lives. But the best lesson we learn is from a bad decision we make. At that crucial point in time some people decide to follow the path of learning through their own experiences; while others prefer that a guide or a teacher could help them live a smooth life. Time and experience can be excellent teachers when you actually learn a lesson from your poor decisions. Experience comes from our way of living, understanding and the adjustments we make. It also comes from suffering, agony and the ordeals we are afflicted by. As one of the famous writer puts it together: “Good judgment comes from experience and experience comes from poor judgment.” Growth starts as soon as you recognize your mistake and the way to prevent it from happening again. Every human being is bound to make mistakes in life, this is normal; but how you learn from them is really the important factor. The only way to prevent oneself from making a mistake the second time is to learn. If you don’t, you will be making that same error again and again until you are forced to learn. I’ll give you an example of my own life. I started playing Table Tennis in 2008. Back then our school had recently bought the equipment for this game and there were only few good players of this game in the whole school.......

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...There is no way like the American way What does the slogan actually mean? An American would probably say that the American way of living is the best way of living and with high standards. But is it how the Americans really live? Fifty years ago, that might have been the answer from a white American, but for a colored American, an African American for instance, it is a whole other story. Back then, around the sixties, there weren’t much you could call laws that supported the blacks. In fact, they didn’t really have any rights at all. They weren’t allowed to sit on the same benches as the whites, nor the same buses as them. Women weren’t even allowed to vote! The females’ rights to vote happened in the twenties, but only for white women. Thinking that black women aren’t worthy enough to vote is absurd. That is what a country is about though, right? It is the people who are to decide how they want their country to be formed - All of them. Not just men. Not just the whites. They can’t just decide what other people want. The injustice does not stop there. Even as a child they grew up, believing that the black Americans were less worth than the whites. At school the black and white children went to different schools, as if they were different from each other. White children, and youth too, would refuse to go to the same school as “negroes”. Why and for what reason? Were the white youngsters smarter than the black? I doubt it. Did the blacks have more “problems”......

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