# Complex Polynomial: Show Coefficients Bounded by Max |p(z)|

• de1irious
In summary, the conversation discusses the problem of proving that each coefficient ak in the polynomial p(z) is bounded by M, given that the maximum absolute value of p(z) for |z|=1 is also M. The individual is trying to solve the problem by taking derivatives, but is stuck. The suggestion of using Cauchy's formula is mentioned as a potential solution.
de1irious
If p(z)=a0+a1z+...+anz^n, and max|p(z)|=M for |z|=1, show that each coefficient ak is bounded by M.

I'm trying to take derivatives but it's not getting anywhere. Thanks!

de1irious said:
If p(z)=a0+a1z+...+anz^n, and max|p(z)|=M for |z|=1, show that each coefficient ak is bounded by M.

I'm trying to take derivatives but it's not getting anywhere. Thanks!

Did you replace z with:

$$exp(i \theta)$$

I suppose differentiating with respect to theata would make sence:

This would give you:

$$\sum_0^Na_n(-i)^n exp(-i n \theta)=\sum_0^Na_n exp(i n (\theta+\pi/2))=\sum_0^Na_n (i Z)^n$$

Yeah, I'm stuck to.

Last edited:
does cauchy's formula help?

Last edited:

## 1. What is a complex polynomial?

A complex polynomial is a mathematical function that involves variables raised to different powers and can contain complex numbers as coefficients. It is often written in the form of p(z) = anzn + an-1zn-1 + ... + a1z + a0, where an, an-1, ..., a1, a0 are complex numbers and z is a variable.

## 2. What does it mean for coefficients to be bounded by max |p(z)|?

Bounded coefficients in a complex polynomial refer to the concept of a maximum limit. In other words, it means that the coefficients of the polynomial are limited or confined to a certain range of values, as determined by the maximum magnitude of the polynomial's overall value. This helps to define the behavior and properties of the polynomial, such as its roots and degree.

## 3. How do you determine the maximum magnitude of a complex polynomial?

The maximum magnitude of a complex polynomial can be found by evaluating the polynomial at different values of z and then taking the absolute value of the resulting number. The largest value obtained in this way is the maximum magnitude.

## 4. Why is it important for coefficients to be bounded by max |p(z)|?

Knowing the maximum magnitude of a complex polynomial's coefficients can provide valuable information about the polynomial's behavior and properties. It can help determine the degree of the polynomial and its roots, and it can also be used in various mathematical calculations and analyses. Additionally, bounded coefficients can make it easier to analyze and solve complex polynomial equations.

## 5. How can you use the concept of bounded coefficients in real-world applications?

The concept of bounded coefficients in complex polynomials has many practical applications in fields such as engineering, physics, and economics. It is used in modeling and predicting various phenomena, such as the behavior of electric circuits, the motion of particles, and the growth of populations. It is also used in optimization problems, where finding the maximum or minimum value of a function is essential.

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