# Complex analysis and linear algebra

Exercise 7. The family of mappings introduced here plays an important role in complex analysis. These mappings, sometimes called Blaschke factors, will reappear in various applications in later chapters.

(1) Let z, w be two complex numbers such that zw 6= 1. Prove that w − z 1 − wz < 1 if |z| < 1 and |w| < 1, and also that w − z 1 − wz = 1 if |z| = 1 or |w| = 1.

(2) Prove that for a fixed w in the unit disc D, the mapping F : z 7→ w − z 1 − wz satisfies the following conditions

(a) F maps the unit disc to itself (that is, F : D → D), and is holomorphic.

(b) F interchanges 0 and w, namely F(0) = w and F(w) = 0.

(c) |F(z)| = 1 if |z| = 1.

(d) F : D → D is bijective.

## Calculate the price of your order

550 words
We'll send you the first draft for approval by September 11, 2018 at 10:52 AM
Total price:
\$26
The price is based on these factors:
Number of pages
Urgency
Basic features
• Free title page and bibliography
• Unlimited revisions
• Plagiarism-free guarantee
• Money-back guarantee
On-demand options
• Writer’s samples
• Part-by-part delivery
• Overnight delivery
• Copies of used sources
Paper format
• 275 words per page
• 12 pt Arial/Times New Roman
• Double line spacing
• Any citation style (APA, MLA, Chicago/Turabian, Harvard)

# Our guarantees

Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

### Money-back guarantee

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

### Zero-plagiarism guarantee

Each paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

### Free-revision policy

Thanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.