Complex variables, maximum value

Thread starter babyrudin
Start date Mar 18, 2008
Tags

Complex Complex variables Maximum Value Variables

In summary, complex variables are quantities that involve both real and imaginary numbers and are used in the study of functions and equations in complex analysis. The maximum value of a complex variable can be determined by finding critical points and evaluating them using the second derivative test. Complex variables differ from real variables in that they involve both real and imaginary components, allowing for a wider range of mathematical operations. They can have multiple maximum values when there are multiple critical points resulting in the same maximum value. Real-world applications of complex variables and maximum value include engineering, physics, and economics, particularly in calculating efficiency and optimizing processes.

Mar 18, 2008

babyrudin

Homework Equations

Let [tex]w \in C[/tex] be a fixed complex number with [tex]|w|<1[/tex]. Let
[tex]f(z)=\frac{z-w}{1-\bar{w}z}.[/tex]
Calculate its maximum value in the region [tex]|z| \leq 1[/tex].

The Attempt at a Solution

How should I approach this? Not sure if maximum modulus principle is of much help.

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Mar 19, 2008

babyrudin

Any idea on this?

1. What are complex variables?

Complex variables are mathematical quantities that involve both real and imaginary numbers. They are often used in the study of functions and equations in complex analysis.

2. How are maximum values determined in complex variables?

The maximum value of a complex variable can be determined by finding the critical points of the function and evaluating them using the second derivative test. The highest value obtained is the maximum value.

3. How do complex variables differ from real variables?

Complex variables differ from real variables in that they involve both real and imaginary components, while real variables only involve real numbers. This allows for a wider range of mathematical operations and applications in complex analysis.

4. Can complex variables have more than one maximum value?

Yes, complex variables can have multiple maximum values. This can occur when the function has multiple critical points that result in the same maximum value.

5. What are some real-world applications of complex variables and maximum value?

Complex variables and maximum value are used in various fields such as engineering, physics, and economics. They are particularly useful in calculating the maximum efficiency of systems and optimizing processes.