Complex Analysis: Sums of elementary fractions

In summary, complex analysis is a branch of mathematics that deals with functions of complex numbers. Elementary fractions in complex analysis are rational functions expressed as a sum of terms with a single complex variable in the denominator. The main focus of studying these fractions is to understand the behavior and properties of rational functions, which are commonly used to model real-world phenomena. To find sums of elementary fractions, techniques such as partial fraction decomposition, contour integration, and the residue theorem are used in complex analysis. These fractions are also useful in real-life applications, including engineering, physics, and economics, as they can be used to analyze complex systems and provide solutions to problems involving differential equations or optimization.
  • #1
Potage11
7
0
I have a homework question that reads:
Represent the following rational functions as sums of elementary fractions and find the primitive functions ( indefinite integrals );

(a) f(z)=z-2/z^2+1

But my confusion arrises when I read sums of elementary fractions.
I think what the question is asking is, show that it is holomorphic, in order to use the property g'(z)=f(z).

Could someone clarify this maybe?
 
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  • #2
It just means that as a whole, [tex]\int \frac{z-2}{z^2+1} dz[/tex] may be a hard integral, but [tex]\int \frac{z}{z^2+1} dz -2\int \frac{1}{z^2+1} dz[/tex] are 2 easy ones.
 
  • #3


Sure! Complex analysis deals with functions that are defined on complex numbers, rather than just real numbers. In this case, the function f(z) is defined as a rational function, which means it is a ratio of two polynomials in z. In other words, it can be written as a sum of elementary fractions, where each fraction has a polynomial in the numerator and denominator.

Finding the primitive function, or indefinite integral, of f(z) involves finding a function F(z) such that F'(z) = f(z). This can be done by breaking f(z) into its elementary fractions and using the known integration rules for polynomials.

For example, in the given function f(z), we can rewrite it as f(z) = (z-2)/(z^2+1) = (z-2)/(z+i)(z-i). This can be further broken down into two elementary fractions, (z-2)/(z+i) and (z-2)/(z-i).

To find the primitive function, we can use the integration rule for polynomials, which states that the primitive function of a polynomial is equal to the polynomial divided by its highest power term. In this case, the highest power term is (z+i)(z-i), so we can write the primitive function as:

F(z) = (1/2)(z-2)/(z+i) + (1/2)(z-2)/(z-i)

Now, to show that f(z) is holomorphic, we need to show that it is differentiable everywhere in its domain. Since we have found a primitive function F(z), this means that f(z) is differentiable everywhere in its domain, and therefore, it is holomorphic.

In summary, the question is asking you to break down the given rational function into its elementary fractions and find the primitive function using integration rules for polynomials. This will show that the function is holomorphic. Hope this helps clarify!
 

1. What is complex analysis?

Complex analysis is a branch of mathematics that studies functions of complex numbers. It deals with the properties and behavior of these functions, which are defined as functions with complex inputs and outputs.

2. What are elementary fractions in complex analysis?

In complex analysis, elementary fractions refer to rational functions that can be expressed as a sum of terms with a single complex variable in the denominator. These fractions are called elementary because they are the most basic form of complex functions.

3. What is the main focus of studying sums of elementary fractions in complex analysis?

The main focus of studying sums of elementary fractions is to understand the behavior and properties of rational functions. This is important in many areas of mathematics and science, as rational functions are often used to model real-world phenomena.

4. What techniques are used in complex analysis to find sums of elementary fractions?

In complex analysis, there are several techniques that are commonly used to find sums of elementary fractions. These include partial fraction decomposition, contour integration, and the residue theorem.

5. How are sums of elementary fractions useful in real life applications?

Sums of elementary fractions are useful in many real-life applications, such as in engineering, physics, and economics. They can be used to model and analyze complex systems, and can also provide solutions to problems involving differential equations or optimization.

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