Free Essay

No Way

In: People

Submitted By AMIDO
Words 4089
Pages 17
3

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



eit · ieit dt =

z dz =
¯
0

0



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

C



(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

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



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,


z −2 dz =
C



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



0

0

C

e(n+1)it dt =

enit ieit dt = i

z n dz =

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


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



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


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

Similar Documents

Free Essay

10 Ways to Get a Raise

...10 WAYS TO GET A RAISE ONLY HAVE EVER ASKED FOR RAISES 44% WOMAN 48% MEN OF THOSE HERE’S HOW TO COME OUT DID GOT WITH THE BEST SCENARIO SOMETHING 85%TO ASK? BRAVE ENOUGH 1. MAKE THE FIRST OFFER THIS CREATES AN ANCHOR THAT YOUR MANAGER WILL HAVE TO WORK DOWN FROM 2. AVOID ROUND NUMBERS A MORE PRECISE NUMBER PROHIBITS A MANAGER FROM NEGOTIATING TOO LOW 73,600 ??? What about 73,000 ??? 3.ASK FOR A RAISE DURING A PROJECT, NOT AFTER WORKERS ARE PROVEN TO BE PRAISED MORE DURING THEIR ASSISTANCE THAN AFTER THEY HAVE COMPLETED THEIR TASKS OTHER SUBTLE TRICKS... 4. ASK FOR A FAVOR Studies show people grow agreeable after paying someone an initial service. 5. DRESS IN A SIMILAR WAY IT’S BEEN PROVED – WE’RE MORE FRIENDLY TOWARD THOSE LIKE US 6. HAVE THINGS IN COMMON WITH YOUR BOSS GOLF? SQUASH? MUSIC BAND? 7. GET A „YES” SET AN AGREEABLE TONE OF MOOD BEFORE ASKING ABOUT THE RAISE 8. CHANGE A LANGUAGE RAISE SALARY ADJUSTMENT 9. BE SWEET TO AVOID SOUNDING GREEDY, PLACE SUGARY TONES IN YOUR BOSS’S EAR 10. MAKE YOUR BOSS LOOK GOOD IN THE CUTTHROAT WORLD OF WORK, APPEARANCES ARE REALITY TO SUM UP... 1. MAKE THE FIRST OFFER 2. AVOID ROUND NUMBERS 3. ASK FOR A RAISE DURING A PROJECT 4. DRESS IN A SIMILAR WAY 5. HAVE THINGS IN COMMON WITH YOUR BOSS 6. GET A „YES” 7. TRACK YOUR ACCOMPLISHMENTS 8. CHANGE A LANGUAGE 9. BE SWEET 10. MAKE YOUR BOSS LOOK GOOD GOOD LUCK ! ...

Words: 257 - Pages: 2

Free Essay

American Way

...treasure: life, liberty, and freedom. This treasure was and still is the American Dream. Now people from all over the world come to America in search of the same Dream; some even die trying. People were not as materialistic as people are now; they just wanted happiness. As time passed, people became more materialistic and began to take for granted what they were born with. However, the “American Dream” hardly ever turns out like any individuals have anticipated. America is often considered as the “best country” in the world, but behind this façade, many people struggle daily to earn enough money to survive. America has had its times of despair and advances but through it all we have learned new ways, cultures, and overall advancements in life. Throughout the American Ways book we were given to read and analyze, there were many things about American life that not only related to now but also showed us lessons learned and conflicts repeated through time. To myself and maybe others, the pieces presented to me in class had the ability to bring me into another world. In each piece, I was almost warped to a different time of America and had the ability to see many things. One thing I would like to compare is not...

Words: 1179 - Pages: 5

Premium Essay

The Toyota Way

...The Toyota Way I have learned about “the Toyota Way”, which leads Toyota to success. “The Toyota way” is the core value and culture of Toyota. I learned a lot from “the Toyota way”. The following are 5 basic principles of “the Toyota Way”: First, challenge. Challenge means that a corporation should always challenge itself in terms of challenging the process. A company should not say that we are doing very well and we need not make any improvements. Unless a company always challenges the process, it cannot have constant competitive. Secondly, improvement An organization should keep improving itself. As the CEO of Toyota said, there is no perfect car; we are improving our cars base on customer’s needs. We should never think we are the best; we should keep identifying where we can still work on and improve. Consistent improvement is one of the traits of a great company. Thirdly, go and see All managers and employees should walk around to observe and find problems. This is basically a concept of managing by walking around, however, Toyota requires that all the employees have the ability to identify the potential problems. This ability needs a lot of experiences and knowledge. Fourthly, respect We should respect our employees and customers, as well as partners. We respect others and then they are going to respect us. Then we can a very good relationship with employees, customers and partners, which are very good for us. Fifthly, teamwork Teamwork is very important...

Words: 300 - Pages: 2

Premium Essay

Toyota Way

...The family and history of Toyota Way Sakichi Toyoda was born in 1867 and died in 1930. During his life, the inventor developed several devices but the greatest invention was the Toyoda Power Loom. The powered loomed was developed in 1896 equipped with the new weft-breaking automatic stopping device. In 1924, he developed the world’s first automatic loom with a non-stop shuttle-change motion called the Type-G Toyoda Automatic Loom. After the automatic loom, Sakichi rooted the Toyota Production System (TPS), which is the philosophy of "the complete elimination of all waste" fills all aspects of production in pursuit of the most efficient methods. Kiichiro Toyoda, inherited the philosophy of TPS from Sakichi. The Just-in-Time concept was developed by Kiichiro Toyoda, the founder (and second president) of Toyota Motor Corporation. "Just-in-Time" means making "only what is needed, when it is needed, and in the amount needed." Supplying "what is needed, when it is needed, and in the amount needed" according to this production plan can eliminate waste, inconsistencies, and unreasonable requirements, resulting in improved productivity. After years of trial and error with the Just-in-Time concept, he improved the TPS efficiency. For the Just-in-Time system to function, all of the parts that are made and supplied must meet predetermined quality standards called the jidoka process. Jidoka means that a machine safely stops when the normal processing is completed. If quality or equipment...

Words: 305 - Pages: 2

Free Essay

United Way

...United Way Individual Marketing Project July 21, 2015 Leslie Wilkerson Client: United Way Basic Client Need: Increase Donations – Fundraisers, social media, mail, & telephone. Client Service Need: Appeal to the heart of people, if you get to a person’s feelings you will raise more money for a cause than a drive or analytical thinking. Keeping the people informed will increase donations, and make it easy to donate. Clients Target Market: Individuals, Volunteers, Contributors, and local Charities within the communities. Situation Analysis: Goods and Services United Way is the world’s largest privately-funded nonprofit. They are engaging people from all walks of life in nearly 1,800 communities across more than 40 countries and territories worldwide. United Way focus is on education, income and health dealing with children, citizens, and communities. They raise the awareness surrounding key issues that affect everyone, provide financial and one-on-one assistance to those who need it most, and promote social and policy change that helps strengthen people and their communities. Industry Their partners include global, national and local businesses, nonprofits, civic and faith organizations, as well as educators, labor, health organization, government and more. United Way and these organizations together raise more than $5 billion every year to create community solutions that improve life for everyone. The United Way volunteers who give their time, talent or skills...

Words: 1330 - Pages: 6

Premium Essay

United Way

...1. How do you feel, as a potential or actual giver to United Way campaigns, about the “high living” of Aramony? Would these allegations affect your gift giving? Why or Why not? William Aramony, president of United Way who built United Way of America into one of the nation’s premier charities. The United Way has been an umbrella charity that was created as a fundraising organization to support many smaller charities. It has been supported by many business firms by fundraising drives and payroll deductions. He had headed the organization, and under his tenure, the organization grew rapidly, nearly quadrupling donations between 1970 and 1990. On the other hand, he was milking out the organization’s fund for his personal benefits such as significant limousine expenses, high salary and uncontrolled perks, international airfare for himself and guests, personal gifts and luxury items, travelling on the charity's dime for personal reasons, affair with young Florida women, loans and diversions of funds to companies that are owned by family members, a $4 million "golden parachute” etc. When an internal investigation and news reports disclosed his lavish life style, as a potential or actual giver to United Way campaigns, I felt terrible knowing all these fact where my contribution had been misused for his lavish lifestyle. I saw a clear sense of white-collar crime under the opportunity fraud triangle. Charitable organizations depend on contributions that people give freely out of a...

Words: 1172 - Pages: 5

Premium Essay

Milky Way Research Paper

...The center of the Milky Way is a fascinating region, not solely because of its enigmatic characteristics but also because of its complexities. The center of the Milky Way, initially, was described as a “mini-spiral” of hot gas, however as years passed, scientists have realized that this mysterious center was not a spiral, but a distinct, separate point of radio emission corresponding to the exact center of the Galaxy. This point is referred to as “Sagittarius A* or ‘Sgr A*’” and today, Sgr A* is believed to consist of a black hole about 3 times larger than the Sun and has a mass of 2 million times that of the sun.Interestingly, the intensity of x-rays being emitted from this massive hole was much less than expected. Scientists then realized...

Words: 386 - Pages: 2

Free Essay

Ways of Seeing by John Berger

...Ways of Seeing by John Berger On Tuesday, June 5th 2012, I went to the Museum of Fine Arts. The atmosphere was peaceful and quiet even though there were a lot of students from a high school. If you decide to go to a museum you have a special attitude. First of all it is your choice to go there. The museum will not come to you. I spent quite some time walking around to find a picture that was “speaking” to me. Then I came across the “Behind the Scenes: Caring for working of art” room. It is an electrical room with a touch screen under the picture. The screen explains a lot of the different things about the history and background of the painting (picture 1). The painting was of Mrs. Timothy Rogers (Lucy Boylston). I was intrigued by the historical background of the Boylston Family and the timeline (1766 – 1767) during which John Singleton Copley painted six portraits of the Boylston family. It reminded me of a line in John Berger’s essay: “They are declared art when their line of descent can be certified” (150). I wonder how many wonderful and beautiful paintings were just thrown away because they were not of someone important, rich, or famous. There was also information about the cleaning of the painting. I was able to see the difference in the 1921 unclean portrait and the 1950 clean portrait. First, I just saw a nice painting of a woman, but now I can see and appreciate the portrait of Mrs. Timothy Rogers. “The meaning of an image is changed according to what one sees...

Words: 1383 - Pages: 6

Premium Essay

My Way a Discourse Analysis

...My Way by Paul Anka 1969; A Song Lyrics Analysis * Lyrics My Way by Paul Anka 1969 And now, the end is near; And so I face the final curtain. My friend, I'll say it clear, I'll state my case, of which I'm certain. I've lived a life that's full. I've traveled each and every highway; But more, much more than this, I did it my way. Regrets, I've had a few; But then again, too few to mention. I did what I had to do And saw it through without exemption. I planned each charted course; Each careful step along the byway, But more, much more than this, I did it my way. Yes, there were times, I'm sure you knew When I bit off more than I could chew. But through it all, when there was doubt, I ate it up and spit it out. I faced it all and I stood tall; And did it my way. I've loved, I've laughed and cried. I've had my fill; my share of losing. And now, as tears subside, I find it all so amusing. To think I did all that; And may I say - not in a shy way, "No, oh no not me, I did it my way". For what is a man, what has he got? If not himself, then he has naught. To say the things he truly feels; And not the words of one who kneels. The record shows I took the blows - And did it my way! * Theme The lyrics of "My Way" tell the story of a man who, having grown old, reflects on his life as death approaches. He is comfortable with his mortality and takes responsibility for how he dealt with all the challenges of life while...

Words: 1755 - Pages: 8

Premium Essay

Continuous Improvement the Toyota Way

...The “Respect for People” principle is one of the two pillars of The Toyota Way; the other being “Continuous Improvement.” Toyota claims that respect for people is the foundation for continuous improvement. Many managers seem to think they know what this “Respect for People” principle implies, and they believe that they follow it. The reality is that most do not understand this concept outside of the Toyota management system. Toyota states, “Respect for people is the attitude that regards people’s ability to think most.” Most managers have a poor idea of what it actually means to demonstrate “Respect for People.” Many would claim that showing respect for people would include things such as treating employees fairly, giving them clear goals, trusting them to achieve goals set, and listening to employees. Managers believe respect is easy to understand and apply these misguided ideas. This is a huge part of lean that has been missing. “Respect for People” is an aspect of excellence at Toyota that needs to be understood and implemented. It is a mindset that can be difficult to understand without experiencing day to day. This is why it was looked over for so long, with instead the focus being on the surface of the Toyota way. Emphasis is put on the high importance of workers capabilities to begin to describe what it means to truly demonstrate “Respect for People. The workers are allowed to display their capabilities through active participation in running and improving...

Words: 1148 - Pages: 5

Premium Essay

Neo-Liberalism and the Third Way

...Essay Topic One: Explain some of the criticisms of neo liberalism and economic rationalism. How is the third way seen as an advance on neo liberalism and what do critics make of the third way? Neo liberalism and the Third Way Neo liberalism and economic rationalism, synonymous with globalisation, gained prominence during the 1990s as a political ideology that espoused free markets as a method of not only handling affairs of nations, but also as political ideology that can be applied with all sorts of governance issues (Hartman 2005, pp.57-59). The idea was that “market forces would motivate workers and produce a vibrant economy” and that “government interference” through “high taxation and investment in the public sector services, lessened individual freedom and hindered a nations economic progress” (Rees 1994, p.172). Emphasis was placed on costs to the government being shifted back to private individuals and markets (Hartman 2005, p. 59). This would create a situation where markets would flourish with increased competition and reducing the tax burden (Hartman 2005, p.59). This came at the cost of the social state (Rees 1994, pg.171) as countries sought to create “market friendly, liberalized, lean, decentralised, customer orientated, managerial and democratic state[s],” (Bangura 2000, p.17). In Australia the conversion of public services into market commodities and increased privatisation of public provisions (Jimrozik 2005, p.42) found a scapegoat in the welfare state (Rees...

Words: 2881 - Pages: 12

Free Essay

Threat to American Way of Life

...The threat America’s way of life is a quite questionable context to define. What I see here is a Catch 22, where we either allow immigrants and that changes America, or we don’t allow immigrants and that changes America. As John Isbister said, “The essence of American life is that it is composed of different groups, different cultures, races, religions, attitudes, folkways and ideologies, differences that give the country its distinctiveness. Current im-migration is sure to change the mixture, but change is not new; the cultural mixture of America has been changing continuously.” There is no auto defined definition to the American way of life, it is always changing whether we want it to or not. As Charon mentions, “Change occurs in every social organization, and it is ongoing and inevitable”. However, I do see this threat to be almost negligible, our biggest problem is within the racial conflictions of society adapting to a change. America has always been seen as a melting pot of cultures; however the melting pot concept is too vague to define. A melting pot means the society confirms to a new identity, however this is not such the case. The traditions are all varying and different outlooks still; we are not some culture defined as a mix of one pot, but rather a mix of many other pots. This correlates to Charon’s idea of the individual making a difference, “Attempts to influence society are countered by the power of long-standing social patterns that are normally defended by...

Words: 755 - Pages: 4

Premium Essay

One Way Traffic

...University of Phoenix Material Traffic Modeling Figure 1 shows the intersections of five one-way streets and the number of cars that enter each intersection from both directions. For example, I1 shows that 400 cars per hour enter from the top and that 450 cars per hour enter from the left. See the Applications section in Section 6.2 of College Algebra as a reference. For this assignment, use Figure 1 to answer the questions following the figure and to prepare a Microsoft® PowerPoint® presentation. [pic] Figure 1. The intersections of five one-way streets The letters a, b, c, d, e, f, and g represent the number of cars moving between the intersections. To keep the traffic moving smoothly, the number of cars entering the intersection per hour must equal the number of cars leaving per hour. 1. Describe the situation. • In this traffic model the pictures illustrates that as cars go out in one direction there is a number of cars coming that are equivalent to the total number of cars going out. The traffic flows through B, C and d will remain a constant, and traffic that flows through the other intersection will change. 2. Create a system of linear equations using a, b, c, d, e, f, and g that models continually flowing traffic. 3. Solve the system of equations. Variables f and g should turn out to be independent. 4. Answer the following questions: a. List acceptable traffic flows for two different values of the independent...

Words: 334 - Pages: 2

Free Essay

10 Ways It Wastes Money

...10 ways IT wastes money on development Date: June 29th, 2009 Author: Justin James To deliver value, IT departments must keep a tight rein on how they use their budget - and that includes development efforts. Justin James cites some of the most common areas where IT throws its development bucks down the drain. Many IT departments walk a fine line between being a needed cost center, a necessary evil, and an absolute money pit. In a few rare occasions, they’re able to deliver enough value to become profit centers. One of the key factors in determining whether an IT department is delivering ROI has to do with the development efforts occurring within that department. Here are 10 of the most common ways IT departments waste money on development. 1: Communication problems Communication issues are one of the biggest causes of project failure. These issues are magnified with internal projects. Just because the “customer” works in the same building and has their paychecks signed by the same person as you doesn’t mean there won’t be communication problems. In fact, internally facing projects are often worse than projects for paying customers because internal customers don’t have to adhere to a contract, and no tangible value is placed upon the work. In these situations, there is little incentive for the customers to work well with you, and if communication breaks down, they complain about how “IT is stonewalling us.” The result is wasted time and money due to things being delivered...

Words: 1806 - Pages: 8

Premium Essay

A Long Way

...A Long Way Gone Essay Matthew Morgan Prof. Carey “On Democracy” Due: 02/27/08 For the “Everybody Reads” assignment I choose to attend the Central library book group discussion. When I first got there I was really surprised because I thought it was going to be a bigger event than what it was. There was only about 10 people total, and 5 of us were students who were there for this exact assignment. It was a really interesting discussion because half of the people that attended were my age and the other half was about two generations older, so there was a very diverse pool of perspectives and opinions. But because there was a large generation gap it was a bit more difficult for me to share my views, so I mainly listened and observed other people’s thoughts. The discussion itself was very helpful because of the different views people had about the memoir. One of the themes of A Long Way Gone that we discussed was the importance of hope. We mainly talked about how this theme was not constant throughout the memoir and that it changed with time. For example one person brought up how at first Ishmael’s only motivator was the hope of his parents being alive, then when he realized that he would never be reunited with them he had lost his hope. It was only when he remembered what his father had said about a person only lives if they have something to live for which gave him his hope back. As far as themes that’s really the only one that we discussed, but we did discuss a lot...

Words: 1027 - Pages: 5