**Problem 9:**Let \(\Omega\) be a bounded open subset of \(\mathbb{C}\), and \(\varphi : \Omega \to \Omega\) a holomorphic function. Prove that if there exists a point \(z_0 \in \Omega\) such that\[\varphi(z_0) = z_0 \quad \text{and} \quad \varphi'(z_0) = 1\]then \(\varphi\) is linear.**Hint:** Why can one assume that \(z_0 = 0\)? Write \(\varphi(z) = z + a_n z^n + O(z^{n+1})\) near 0, and prove that if \(\varphi_k = \varphi \circ \cdots \circ \varphi\) (where \(\varphi\) appears \(k\) times), then \(\varphi_k(z) = z + k a_n z^n + O(z^{n+1})\). Apply the Cauchy inequalities and let \(k \to \infty\) to conclude the proof. Here we use the standard \(O\) notation, where \(f(z) = O(g(z))\) as \(z \to 0\) means that \(|f(z)| \leq C|g(z)|\) for some constant \(C\) as \(|z| \to 0\).

Answered: Image /qna-images/answer/62b63449-4303-4b61-8830-8254655a53b2.jpg

Answered: 9. Let be a bounded open subset of C,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Prove the theorem
5). Let 21, 22 be fixed elements of a Banach space X, and l1, 2 E X'. Define A: X→ X by Ax = l₁(x)2₁ + 2(x) 22. Show that A is compact.
1. Find curl(v) of the following in terms of (x1, x2, x3) and (e1, e2, e3). If curl(v) = 0, find the function f such that Vf = v. a. v(x) = (x² + x + x3)-'(x1e1 + x2€2 + X3@3). b. v(x) = (xỉ + X2X3)e1 + (x3 + X1X3)e2 + (xž + x2X1)e3 c. v(x) = e"1(sin(x2) cos(x3)e1 + sin(x2) sin(x3)e2 + cos(x2)e3)
2. Let [0, 1] := {ƒ : [0,1] → R : f is a bounded function on [0, 1]}. Let ||f|| sup f(x)| x= [0,1] for fel [0, 1]. Show that ( [0, 1], || - ||) is a Banach space.
Let T: R³ R2 be a linear mapping. → 10 *([:]) = [ ] ¹ ({]) - [ ] -~-~(:)) - [²7] ([]) - T = and T find T (ED). Given that T sin (a) Ər f α Ω E
2. Suppose f is holomorphic in an open set UCC and a E U. Show that for any integer n > 0 there is a unique holomorphic function h:U → C such that f(z) = f(a)+f'(a)(z-a)+ f"(a)²- a)² 2! (z – a)" +f(m) (a). | +(z-a)"+'h(z), ze U. n! n+1
5. Consider the function f! NXN → N, defined by f(m,n) = min +1 a) find f(A)= A= {(1,0), (0, 1), (1,1) 3 b) find pre-image of B = {0,13 by f 1.e., F-1 (B) := f `¹ (0) U f `` (1) c) Is f an injective (or one-to-one) function.? Justify answer. d) Is f a surjective (or onto) function. Justify answer.
Verify the following result. Theorem. Let U be an open subset of R" and let f:U → R be a C' function Then if ro E U is either a local marimum or a local minimum of f subject to the constraints 0 = (x)*6 for i e {1,..., m} (the g, being differentiable), then there exist scalars A1,..,Am and a scalar p, not all zero, such that ( fz, (x0)\ 9kz1 (T0) (1) \Sz, (T0)/ \9kIn (T0) If the vectors 9izı (20) Fmz, (T0) \9iz, (T0), are linearly independent, it follows that an equation of the form \Imzn (T0). 9kz; (x0)) (2) \Szn (#0)/ k=1 holds. (02) "z¥6\ Hint: Use the implicit function theorem.
6. If af/dx is continuous on the rectangle R = {(t, æ) : 0 0 such that |f(t, x1) – f(t, x2)| < K\x1 – x2| for all (t, æ1) and (t, x2) in R. 7. Define the sequence {un} by uo (t) = 20, Un+1 = x0 + to + f(s, un(8)) ds, п %3 1,2,.... Use the result of the previous exercise to show that |f(t, un (t)) – f(t, un-1 (t))| < K|u,(t) – Un-1(t)|.
3. (.) Denote the number of derangements of {1, 2,..., n} by Dn. Using the formula for the number of the derangement that we found in class, provide a proof of the relation D, = (n – 1)(D-2 + Dn-1); (n = 3,4, 5, ...).

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,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS