Consider differentiable functions a(t), B(1), a, B: [lo, l₁] → R, and set µ(t) = a(t)+iß(t). Explain why rt₁ [{"^ p² (1) dt = µ(tr) — µ(lo). to [You may use the Fundamental Theorem of Calculus for real-valued functions in a real variable. The question is asking you to deduce a version for complex functions.] Now let f: U → C be a holomorphic function and y: [lo, t₁] → U a differentiable path from a € U to be U. Suppose that f(z) = g'(z) where g: U →→ C is another holomorphic function: that is, f has an antiderivative g on U. Show the Complex Fundamental Theorem of Calculus, [ f(z) dz = f*^* f(y(1))7′(1) dl = g(b) – g(a). [You can assume the first equality holds; the question is about the second.] [Hint: set u(t)=h(y(1)). What is '(t)? What does (a) give?]
Consider differentiable functions a(t), B(1), a, B: [lo, l₁] → R, and set µ(t) = a(t)+iß(t). Explain why rt₁ [{"^ p² (1) dt = µ(tr) — µ(lo). to [You may use the Fundamental Theorem of Calculus for real-valued functions in a real variable. The question is asking you to deduce a version for complex functions.] Now let f: U → C be a holomorphic function and y: [lo, t₁] → U a differentiable path from a € U to be U. Suppose that f(z) = g'(z) where g: U →→ C is another holomorphic function: that is, f has an antiderivative g on U. Show the Complex Fundamental Theorem of Calculus, [ f(z) dz = f*^* f(y(1))7′(1) dl = g(b) – g(a). [You can assume the first equality holds; the question is about the second.] [Hint: set u(t)=h(y(1)). What is '(t)? What does (a) give?]
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
q1
![1. (a) Consider differentiable functions a(t), B(1), a, B: [lo, 1] → R, and set u(t)= a(l) +iB(1). Explain
why
rt₁
[^^ µ² (1) dt = µ(tr) — µ(to).
[You may use the Fundamental Theorem of Calculus for real-valued functions in a real variable.
The question is asking you to deduce a version for complex functions.]
(b) Now let f: U → C be a holomorphic function and y: [lo, t₁] → U a differentiable path from a € U
to be U. Suppose that f(z) = g'(z) where g: U → C is another holomorphic function: that is, f
has an antiderivative g on U.
Show the Complex Fundamental Theorem of Calculus,
-t₁
[₁1(2)
(z) dz = √² )
=
f*^* f(x(0))Y' (1) di = g(b) — 9(a).
to
[You can assume the first equality holds; the question is about the second.]
[Hint: set u(t)=h(y(1)). What is µ'(t)? What does (a) give?]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb7d0892c-a19b-49d0-ac25-73d2c42d4d77%2F711da492-469c-4cf5-a76a-ff592715a74b%2Fylo3msq_processed.png&w=3840&q=75)
Transcribed Image Text:1. (a) Consider differentiable functions a(t), B(1), a, B: [lo, 1] → R, and set u(t)= a(l) +iB(1). Explain
why
rt₁
[^^ µ² (1) dt = µ(tr) — µ(to).
[You may use the Fundamental Theorem of Calculus for real-valued functions in a real variable.
The question is asking you to deduce a version for complex functions.]
(b) Now let f: U → C be a holomorphic function and y: [lo, t₁] → U a differentiable path from a € U
to be U. Suppose that f(z) = g'(z) where g: U → C is another holomorphic function: that is, f
has an antiderivative g on U.
Show the Complex Fundamental Theorem of Calculus,
-t₁
[₁1(2)
(z) dz = √² )
=
f*^* f(x(0))Y' (1) di = g(b) — 9(a).
to
[You can assume the first equality holds; the question is about the second.]
[Hint: set u(t)=h(y(1)). What is µ'(t)? What does (a) give?]
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 with 3 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,
