8. Use the method in Example 2, Sec. 19, to show that f'(z) does not exist at any point z when (a) f(z) = Rez; (b) f(z) = Im z.
8. Use the method in Example 2, Sec. 19, to show that f'(z) does not exist at any point z when (a) f(z) = Rez; (b) f(z) = Im z.
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
Transcribed Image Text:8. Use the method in Example 2, Sec. 19, to show that f'(z) does not exist at any point
z when
(a) f(z) = Rez;
(b) f(z) = Im z.
![ot
2
:)
SEC. 19
(0, 0) horizontally through the points (Δx, 0) on the real axis (Fig. 28),
Δz = Δx + 10 = Δx - i0 = Δx + i0 = Δz.
In that case, expression (4) tells us that
Δη
Δω
Δε Δε
Hence if the limit of Aw/Az exists, its value must be unity. However, when Az
approaches (0, 0) vertically through the points (0, Ay) on the imaginary axis, so that
Δz = 0 + i Δy = 0 - i4y = -(0 + ίΔy) = − Δz,
we find from expression (4) that
Δy]
(0, Ay)•
Δω -Δz
Δε | Δε
Hence the limit must be -1 if it exists. Since limits are unique (Sec. 15), it follows
that dw/dz does not exist anywhere.
(0, 0)
(Δx, 0)
= 1.
= -1.
FIGURE 28
EXAMPLE 3. Consider the real-valued function f(z) = |z|2. Here
Δω
|z+Az|² - 12/²
Δε
(z + Δz)(z + Δz) – 27
Δε
Δε
and since z + Az = z+Az, this becomes
(5)
respectively, we have the expressions
Δω
ΔΕ
=ā + Δz + z
DERIVATIVES 57
Δω
Δε
Proceeding as in Example 2, where horizontal and vertical approaches of Az toward
the origin gave us
Δε
Δε
Δz = Δz and Δz = -Δz,
=ā+Δz+z when Δz = (Δ.x, 0)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6389447-1237-4af0-b5c6-eb1260425b55%2F68903a62-0aba-48b3-87ae-5a57004abd10%2Fo6uz3kq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:ot
2
:)
SEC. 19
(0, 0) horizontally through the points (Δx, 0) on the real axis (Fig. 28),
Δz = Δx + 10 = Δx - i0 = Δx + i0 = Δz.
In that case, expression (4) tells us that
Δη
Δω
Δε Δε
Hence if the limit of Aw/Az exists, its value must be unity. However, when Az
approaches (0, 0) vertically through the points (0, Ay) on the imaginary axis, so that
Δz = 0 + i Δy = 0 - i4y = -(0 + ίΔy) = − Δz,
we find from expression (4) that
Δy]
(0, Ay)•
Δω -Δz
Δε | Δε
Hence the limit must be -1 if it exists. Since limits are unique (Sec. 15), it follows
that dw/dz does not exist anywhere.
(0, 0)
(Δx, 0)
= 1.
= -1.
FIGURE 28
EXAMPLE 3. Consider the real-valued function f(z) = |z|2. Here
Δω
|z+Az|² - 12/²
Δε
(z + Δz)(z + Δz) – 27
Δε
Δε
and since z + Az = z+Az, this becomes
(5)
respectively, we have the expressions
Δω
ΔΕ
=ā + Δz + z
DERIVATIVES 57
Δω
Δε
Proceeding as in Example 2, where horizontal and vertical approaches of Az toward
the origin gave us
Δε
Δε
Δz = Δz and Δz = -Δz,
=ā+Δz+z when Δz = (Δ.x, 0)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 5 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
