# Proof that e^z is not a finite polynomial

• freefall111
In summary: says that a polynomial of degree n has (n+1)st derivative = 0 identically, so that looks like the case here.
freefall111

## Homework Statement

Prove that the analytic function e^z is not a polynomial (of finite degree) in the complex variable z.

## The Attempt at a Solution

The gist of what I have so far is suppose it was a finite polynomial then by the fundamental theorem of algebra it must have at least one or more roots. e^z can never equal zero for hence this is a contradiction.

Is it okay for me to apply the fundamental theorem of algebra like this or am I kind of using a bit too much machinery here?

Your proof is wrong. Indeed: $e^{2\pi i}=0$, so the function DOES have a root!

micromass said:
Your proof is wrong. Indeed: $e^{2\pi i}=0$, so the function DOES have a root!
Wait, what? $e^{i2\pi} = 1$

Woow, I'm stupid today.

I'm sorry, your proof is alright! I obviously need to get some sleep.

freefall111 said:

## Homework Statement

Prove that the analytic function e^z is not a polynomial (of finite degree) in the complex variable z.

## The Attempt at a Solution

The gist of what I have so far is suppose it was a finite polynomial then by the fundamental theorem of algebra it must have at least one or more roots. e^z can never equal zero for hence this is a contradiction.

Is it okay for me to apply the fundamental theorem of algebra like this or am I kind of using a bit too much machinery here?

That looks like too much machinery (at least for my tastes); I would rather just argue that a polynomial of degree n has (n+1)st derivative = 0 identically, and ask whether that can happen for exp(z).

RGV

## What is e^z?

e^z is the mathematical constant known as Euler's number, approximately equal to 2.71828. It is the base of the natural logarithm and has many applications in mathematics and science.

## What is a finite polynomial?

A finite polynomial is a mathematical expression made up of a finite number of terms, each consisting of a constant coefficient and a variable raised to a non-negative integer power. Examples include x^2 + 3x + 5 and 2x^3 + 8x^2 - 4x + 1.

## Why is e^z not a finite polynomial?

e^z cannot be written as a finite polynomial because it has an infinite number of terms. Its Taylor series expansion includes terms of all powers of z, starting with z^0 (which is equal to 1) and continuing to infinity. This makes it impossible to represent e^z as a polynomial with a finite number of terms.

## What is the difference between e^z and a finite polynomial?

The main difference between e^z and a finite polynomial is that e^z has an infinite number of terms, while a finite polynomial has a finite number of terms. This means that e^z is not a polynomial, as it does not fit the definition of a polynomial.

## What are the implications of e^z not being a finite polynomial?

The fact that e^z is not a finite polynomial has important consequences in mathematics and science. It means that e^z cannot be manipulated using the rules and properties of finite polynomials, and it requires a different approach to be studied and analyzed. Additionally, it is an essential concept in calculus and complex analysis, and its properties are used in various areas of science and engineering.

• Calculus and Beyond Homework Help
Replies
24
Views
2K
• Calculus and Beyond Homework Help
Replies
6
Views
2K
• Calculus and Beyond Homework Help
Replies
7
Views
497
• Calculus and Beyond Homework Help
Replies
15
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
2K
• Linear and Abstract Algebra
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
7
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus
Replies
3
Views
469