Given a E R, let fa(2) be the a-branch of the complex square root, that is: for every z E C set fa(2) = Vr e' arga/ , where arga(z) is the unique angle 0 € (a, a + 27] such that z = r e?". (i) Show that fr/4(2i) = 1+ i and f3/4(2i) = -(1+ i).
Given a E R, let fa(2) be the a-branch of the complex square root, that is: for every z E C set fa(2) = Vr e' arga/ , where arga(z) is the unique angle 0 € (a, a + 27] such that z = r e?". (i) Show that fr/4(2i) = 1+ i and f3/4(2i) = -(1+ i).
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
Topic Video
Question
100%
Could you explain how to show this in detail?
![Given a E R, let fa(z) be the a-branch of the complex square root, that is: for every z E C
set
fa(2) = Vreiarga/2 .
where arga(z) is the unique angle 0 € (a, a + 27] such that z = r
6.
(i) Show that fm/4(2i) = 1+ i and f37/4(2i)
-(1+i).
{z E C : z =
tea , t > 0} show that fa(z) is not continuous at any
(ii) Setting La
z E La.
(iii) Show that fa(z) = U(r,0) + iV(r,0) with
U (r, 0) = Vr cos
V (r, 0) = /r sin
G).
for r > 0 and a < 0 < a+2ñ, and thus that fa(z) is holomorphic on A = C\La with
1
fa(2) =
Vz E A.
2 fa(2) '](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2Fbf5d4f24-4ec1-40c1-a3c1-f85b4f6643d8%2F2vp2hr_processed.png&w=3840&q=75)
Transcribed Image Text:Given a E R, let fa(z) be the a-branch of the complex square root, that is: for every z E C
set
fa(2) = Vreiarga/2 .
where arga(z) is the unique angle 0 € (a, a + 27] such that z = r
6.
(i) Show that fm/4(2i) = 1+ i and f37/4(2i)
-(1+i).
{z E C : z =
tea , t > 0} show that fa(z) is not continuous at any
(ii) Setting La
z E La.
(iii) Show that fa(z) = U(r,0) + iV(r,0) with
U (r, 0) = Vr cos
V (r, 0) = /r sin
G).
for r > 0 and a < 0 < a+2ñ, and thus that fa(z) is holomorphic on A = C\La with
1
fa(2) =
Vz E A.
2 fa(2) '
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 3 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.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,
