4. Let -{(x4/35/3 + 1x5/3y4/3)/(x² + y²) if z #0 z = 0. Vy= f(z) = Show that the Cauchy-Riemann equations hold at z = 0 but that f is not differen- tiable at this point. [HINT: Consider the difference quotient f(Az)/Az for Az → 0 along the real axis and along the line y = = x.] violivians to armet ni nonny ad 2) = (1,2)) nota 5. Show that the function f(z) = ex²-y² [cos(2xy) + i sin(2xy)] is entire, and find its derivative.cond-order Fx 1x = 6· form to semi- ди 6. If u and v are expressed in terms of polar coordinates (r, e), show that the Cauchy-y: 2
4. Let -{(x4/35/3 + 1x5/3y4/3)/(x² + y²) if z #0 z = 0. Vy= f(z) = Show that the Cauchy-Riemann equations hold at z = 0 but that f is not differen- tiable at this point. [HINT: Consider the difference quotient f(Az)/Az for Az → 0 along the real axis and along the line y = = x.] violivians to armet ni nonny ad 2) = (1,2)) nota 5. Show that the function f(z) = ex²-y² [cos(2xy) + i sin(2xy)] is entire, and find its derivative.cond-order Fx 1x = 6· form to semi- ди 6. If u and v are expressed in terms of polar coordinates (r, e), show that the Cauchy-y: 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
question 5
![2.4 The Cauchy-Riemann Equations
Using Theorem 6 and the Cauchy-Riemann entrations, one can further show that
an analytic function (2) must be constant when any one of the following conditions
hold in a domain D:
The proofs are left as problems.
Th
f(z) =
EXERCISES 2.4
1. Use the Cauchy-Riemann equations to show that the following functions are no-
where differentiable.
(a) w = z
(b) w = Rez
(c) w = 2y - ix
2. Show that h(z)
-
x³ + 3xy² - 3x + i(y³ + 3x²y - 3y) is differentiable on the
coordinate axes but is nowhere analytic.
{
Re f(z) is constant
Im f(z) is constant:
f(z)] is constant.
3. Use Theorem 5 to show that g(z) = 3x² + 2x - 3y2 - 1+i(6xy + 2y) is entire.
Write this function in terms of z.
4. Let
(x4/3y5/3 + ix5/3y4/3)/(x² + y²)
0
-V
4²_vZ
v2
6
& = 1
if z # 0,
if z = 0.
hese han ди
du
ər
01
Show that the Cauchy-Riemann equations hold at z = 0 but that f is not differen-
tiable at this point. [HINT: Consider the difference quotient f(Az)/Az for Az → 0
along the real axis and along the line y = x.]
olivians lo noit
10 armel
5. Show that the function f(z) = ex²-y² [cos(2xy) + i sin(2xy)] is entire, and find its
37
derivative.
Ux6 =3x² +31²
uy = 3y² +3
Vy = 6x+
√x=6
6. If u and v are expressed in terms of polar coordinates (r, 0), show that the Cauchy-
Riemann equations can be written in the form
1 ου dvere 1 du imaginary pens
onal Sch gruoubung
r de orent Ər
r 20
24 = 2
du
y
[HINT: Consider the difference quotient (f(z)-f(zo))/(z-zo), as z → zo= roeio
along the ray arg z = 0o and along the circle |z| = ro.]
Show that if two analytic functions f and g have the same derivative throughout a
domain D, then they differ only by an additive constant. [HINT: Consider f- g.]
Show that if f is analytic in a domain D and either Re f(z) or Im f(z) is constant
n D, then f(z) must be constant in D.
how, by contradiction, that the function F(z) = 2² - z| is nowhere analytic be-
ause of condition (7).
lbsmed at "hed-s
72²-y²= 1274
-2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e554a09-0f9c-418e-bf2f-f101c15adca6%2F0e09ceb4-d0f0-48b8-9cc5-86120d953fc5%2Fwpqev7t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2.4 The Cauchy-Riemann Equations
Using Theorem 6 and the Cauchy-Riemann entrations, one can further show that
an analytic function (2) must be constant when any one of the following conditions
hold in a domain D:
The proofs are left as problems.
Th
f(z) =
EXERCISES 2.4
1. Use the Cauchy-Riemann equations to show that the following functions are no-
where differentiable.
(a) w = z
(b) w = Rez
(c) w = 2y - ix
2. Show that h(z)
-
x³ + 3xy² - 3x + i(y³ + 3x²y - 3y) is differentiable on the
coordinate axes but is nowhere analytic.
{
Re f(z) is constant
Im f(z) is constant:
f(z)] is constant.
3. Use Theorem 5 to show that g(z) = 3x² + 2x - 3y2 - 1+i(6xy + 2y) is entire.
Write this function in terms of z.
4. Let
(x4/3y5/3 + ix5/3y4/3)/(x² + y²)
0
-V
4²_vZ
v2
6
& = 1
if z # 0,
if z = 0.
hese han ди
du
ər
01
Show that the Cauchy-Riemann equations hold at z = 0 but that f is not differen-
tiable at this point. [HINT: Consider the difference quotient f(Az)/Az for Az → 0
along the real axis and along the line y = x.]
olivians lo noit
10 armel
5. Show that the function f(z) = ex²-y² [cos(2xy) + i sin(2xy)] is entire, and find its
37
derivative.
Ux6 =3x² +31²
uy = 3y² +3
Vy = 6x+
√x=6
6. If u and v are expressed in terms of polar coordinates (r, 0), show that the Cauchy-
Riemann equations can be written in the form
1 ου dvere 1 du imaginary pens
onal Sch gruoubung
r de orent Ər
r 20
24 = 2
du
y
[HINT: Consider the difference quotient (f(z)-f(zo))/(z-zo), as z → zo= roeio
along the ray arg z = 0o and along the circle |z| = ro.]
Show that if two analytic functions f and g have the same derivative throughout a
domain D, then they differ only by an additive constant. [HINT: Consider f- g.]
Show that if f is analytic in a domain D and either Re f(z) or Im f(z) is constant
n D, then f(z) must be constant in D.
how, by contradiction, that the function F(z) = 2² - z| is nowhere analytic be-
ause of condition (7).
lbsmed at "hed-s
72²-y²= 1274
-2
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