(b) Find lim f(z) as :→→ 0 along the line y = 2x. (c) Find lim f(z) as z → 0 along the parabola y = x². (d) What can you conclude about the limit of f(z) as z → 0? 11. Let f(z) = /z. Show that f(z) does not have a limit as : → 0. 12. Does u(x, y) = (x²¹ - 3xy²)/(x² + y) have a limit as (x. y) → (0, 0)? 13. Let f(2)= 2¹/² = r/2[cos(8/2) + i sin(0/2)], where r> 0 and -< polar form of z and show that (a) f(z) → i as: -1 along the upper semicircle r = 1,0 <0 0 and - < 0 ≤. Show that f(z) is discontinuous at each point along the negative x axis. 22. Let f(2)= In |z| + i Argz. where - < Arg: ≤ 1. Show that f(z) is discontinuous at 30 = 0 and at each point along the negative x axis. 23. Let A and B be complex constants. Use Theorem 2.1 to prove that lim (Az + B) = A + B.
(b) Find lim f(z) as :→→ 0 along the line y = 2x. (c) Find lim f(z) as z → 0 along the parabola y = x². (d) What can you conclude about the limit of f(z) as z → 0? 11. Let f(z) = /z. Show that f(z) does not have a limit as : → 0. 12. Does u(x, y) = (x²¹ - 3xy²)/(x² + y) have a limit as (x. y) → (0, 0)? 13. Let f(2)= 2¹/² = r/2[cos(8/2) + i sin(0/2)], where r> 0 and -< polar form of z and show that (a) f(z) → i as: -1 along the upper semicircle r = 1,0 <0 0 and - < 0 ≤. Show that f(z) is discontinuous at each point along the negative x axis. 22. Let f(2)= In |z| + i Argz. where - < Arg: ≤ 1. Show that f(z) is discontinuous at 30 = 0 and at each point along the negative x axis. 23. Let A and B be complex constants. Use Theorem 2.1 to prove that lim (Az + B) = A + B.
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 14 and 19 from the Complex Analysis for Mathematics and Engineering book
![(b) Find lim f(z) as z →→ 0 along the line y = 2x.
4)
(c) Find lim f(z) as z → 0 along the parabola y = r².
--4)
(d) What can you conclude about the limit of f(z) as : → 0?
11. Let f(z) = z. Show that f(z) does not have a limit as :→0.
12. Does u(x, y) = (x¹ - 3xy)/(x² + y) have a limit as (r. y) → (0.0)?
Let f(3) = 2/2 = r2[cos(8/2) + i sin(0/2)], where r> 0 and - <0
polar form of z and show that
13.
(a) f(z) →→→ i as :
-1 along the upper semicircle r = 1,0 <0 <n.
(b) f(z) →→→-i asz-1 along the lower semicircle r = 1, -*<O<O
14. Does lim Arg z exist? Why? Hint: Use polar coordinates and let z approach - 4 from
IL
the upper and lower half planes.
15. Determine where the following functions are continuous.
(a) z¹ - 9z²+iz - 2
(b)
z + 1
²+2:+2
2²+62 +5
z + 3x + 2
x + iy
(d)
(e)
1³1 - 1
16. Let f(2)= [z Re(z)/|| when z = 0, and let f(0) = 0. Show that f(z) is continuous for
all values of z.
Use the
(c)
17. Let f(x) = xe +ive. Show that fiz) is continuous for all values of z.
18. Let f(2)=(x² + y²
at z = 0.
when :0. and let f(0) = 1. Show that f(z) is not continuous
19. Let f(2)= Re(z) |z| when : 0, and let f(0) = 1. Is f(z) continuous at the origin?
20. Let f(x)= [Re()/|z| when z0. and let f(0) = 1. Is f(z) continuous at the origin?
21. Let f(z) = 1/2 = [cos(0/2) + i sin(0/2)], where r> 0 and - < 0 ≤. Show that
f(z) is discontinuous at each point along the negative x axis.
22. Let f(2)= In |z| + i Argz. where - < Arg: ≤ 1. Show that f(z) is discontinuous at
30 = 0 and at each point along the negative x axis.
23. Let A and B be complex constants. Use Theorem 2.1 to prove that
lim (Az + B) = A + B.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0faa0630-92f0-4653-b84e-0d22ebfc40eb%2Fb8aeaeee-e78a-4531-8dee-3068501badeb%2Fo5g73u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(b) Find lim f(z) as z →→ 0 along the line y = 2x.
4)
(c) Find lim f(z) as z → 0 along the parabola y = r².
--4)
(d) What can you conclude about the limit of f(z) as : → 0?
11. Let f(z) = z. Show that f(z) does not have a limit as :→0.
12. Does u(x, y) = (x¹ - 3xy)/(x² + y) have a limit as (r. y) → (0.0)?
Let f(3) = 2/2 = r2[cos(8/2) + i sin(0/2)], where r> 0 and - <0
polar form of z and show that
13.
(a) f(z) →→→ i as :
-1 along the upper semicircle r = 1,0 <0 <n.
(b) f(z) →→→-i asz-1 along the lower semicircle r = 1, -*<O<O
14. Does lim Arg z exist? Why? Hint: Use polar coordinates and let z approach - 4 from
IL
the upper and lower half planes.
15. Determine where the following functions are continuous.
(a) z¹ - 9z²+iz - 2
(b)
z + 1
²+2:+2
2²+62 +5
z + 3x + 2
x + iy
(d)
(e)
1³1 - 1
16. Let f(2)= [z Re(z)/|| when z = 0, and let f(0) = 0. Show that f(z) is continuous for
all values of z.
Use the
(c)
17. Let f(x) = xe +ive. Show that fiz) is continuous for all values of z.
18. Let f(2)=(x² + y²
at z = 0.
when :0. and let f(0) = 1. Show that f(z) is not continuous
19. Let f(2)= Re(z) |z| when : 0, and let f(0) = 1. Is f(z) continuous at the origin?
20. Let f(x)= [Re()/|z| when z0. and let f(0) = 1. Is f(z) continuous at the origin?
21. Let f(z) = 1/2 = [cos(0/2) + i sin(0/2)], where r> 0 and - < 0 ≤. Show that
f(z) is discontinuous at each point along the negative x axis.
22. Let f(2)= In |z| + i Argz. where - < Arg: ≤ 1. Show that f(z) is discontinuous at
30 = 0 and at each point along the negative x axis.
23. Let A and B be complex constants. Use Theorem 2.1 to prove that
lim (Az + B) = A + B.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 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,
