Problem 9 - Complex Analysis

Question

Show that every rational function \(R\) (i.e. \(R(z)=P(z) / Q(z), P, Q\) polynomials, \(Q \neq 0)\) can be written as the sum of a polynomial and a finite linear combination, with complex coefficients, of "simple functions" of the form $$ z \mapsto \frac{1}{(z-s)^{n}}, \quad n \in \mathbb{N}, s \in \mathbb{C} $$ the so-called "partial fractions" (Partial fraction decomposition theorem), see also Chapter III, Appendix A to sections III.4 and III.5, Proposition A.7). Deduce: If the coefficients of \(P\) and \(Q\) are real, then \(f\) has "a real partial fraction decomposition" (by putting together pairs of complex conjugate zeros, or rather by putting together the corresponding partial fractions (see also Exercise 10 in I.1).

Step-by-Step Solution

Answer

Write \( R(z) \) as a polynomial plus partial fractions using polynomial division and factorization.

Step 1: Express the Rational Function

Consider a rational function \( R(z) = \frac{P(z)}{Q(z)} \), where both \( P(z) \) and \( Q(z) \) are polynomials and \( Q(z) eq 0 \). The first step in the partial fraction decomposition is to attempt to express \( R(z) \) as a sum of a polynomial and fractions of the form \( \frac{1}{(z-s)^n} \).

Step 2: Identify Partial Fraction Decomposition

Use polynomial division if the degree of \( P \) is greater than or equal to the degree of \( Q \) to express \( R(z) \) as \[ R(z) = S(z) + \frac{R_1(z)}{Q(z)} \]where \( S(z) \) is a polynomial and the degree of \( R_1(z) \) is less than the degree of \( Q(z) \).

Step 3: Factor the Denominator Completely

Once you have \( \frac{R_1(z)}{Q(z)} \), factor \( Q(z) \) completely into linear factors, possibly repeated, i.e., \[ Q(z) = (z - r_1)^{m_1}(z - r_2)^{m_2} \cdots (z - r_k)^{m_k} \]where \( r_i \) are the roots of \( Q(z) \).

Step 4: Set Up the Partial Fractions

Rewrite \( \frac{R_1(z)}{Q(z)} \) as a sum consisting of terms of the form \[ \sum_{i=1}^k \sum_{j=1}^{m_i} \frac{A_{ij}}{(z - r_i)^j} \]where \( A_{ij} \) are constants to be determined. This represents the linear combination of simple functions.

Step 5: Solve for Coefficients

Determine the constants \( A_{ij} \) by multiplying through by \( Q(z) \) and equating coefficients or substituting convenient values of \( z \) into the equation to form a system of equations, which can be solved for each \( A_{ij} \).

Step 6: Consider Real Coefficients

If the coefficients of \( P \) and \( Q \) are real, then the conjugate pair approach is used. This means if \( r \) is a complex root, its conjugate \( \overline{r} \) is also a root. Pair together these partial fractions and express in terms of real partial fractions by combining or rewriting them.

Key Concepts

Rational Functions

A rational function is defined as the ratio of two polynomials, specifically a numerator polynomial over a non-zero denominator polynomial. This type of function is a central object of study in mathematics and plays a significant role in calculus, algebra, and beyond.
Rational functions can take the form of simple fractions in algebra, such as \( R(z) = \frac{P(z)}{Q(z)} \), where both \( P(z) \) and \( Q(z) \) are polynomials and \( Q(z) eq 0 \). Understanding these functions involves exploring their properties, such as domains (where the function is defined) and asymptotic behavior (as values move towards infinity or singular points).

• Domain: The domain of rational functions excludes values that make the denominator equal to zero.
• Asymptotes: Vertical asymptotes occur at values where \( Q(z) = 0 \), and horizontal or oblique ones may describe the end behavior based on the degrees of \( P(z) \) and \( Q(z) \).

Polynomial Long Division

Polynomial long division is an essential tool used to simplify rational functions. It is analogous to the division of numbers but applied to polynomial expressions. This technique helps in breaking down the polynomial expression into simpler parts, which is crucial when considering partial fraction decomposition.
The division process generally follows these steps:

• Arrange the terms of both the dividend (polynomial to divide) and the divisor (the dividing polynomial) in descending order.
• Divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient.
• Multiply the entire divisor by this term and subtract the result from the dividend to find the remainder.
• Repeat the process using the remainder until the remainder has a degree less than the divisor.

This process effectively separates the polynomial part (\( S(z) \)) from the remainder (\( R_1(z) \)) with the degree lower than the initial divisor.

Complex Roots

Complex roots arise when factoring the denominator of a rational function, especially when real polynomial roots are not possible. Complex roots always come in conjugate pairs if the coefficients of polynomials are real.
When performing partial fraction decomposition, recognizing and handling complex roots correctly is crucial. If a denominator polynomial can be factored into linear terms with complex roots, it simplifies to forms like \( (z - r)(z - \overline{r}) \), where \( \overline{r} \) is the conjugate of \( r \).

• Complex Conjugates: If \( r = a + bi \) is a root, then \( \overline{r} = a - bi \) will also be a root.
• Simplifying: The presence of complex conjugates allows us to express partial fractions in simpler terms by combining them, often resulting in terms with real coefficients.

Real Coefficients

When the polynomials \( P(z) \) and \( Q(z) \) in a rational function have real coefficients, it affects the nature of the partial fraction decomposition. Specifically, any complex roots in the factors of \( Q(z) \) will appear in conjugate pairs, allowing for real-valued expressions.
This situation simplifies partial fraction decomposition significantly. By grouping conjugate complex roots together, you can form expressions that ensure the resulting function retains real coefficients throughout its parts.

• Real Partial Fractions: Each pair of complex conjugate roots \((z - r), (z - \overline{r})\) is combined to form terms that sum to real partial fractions.
• Ensuring Real Values: By ensuring that each partial fraction corresponding to a pair of complex conjugates is merged and expressed neatly, real coefficients are maintained throughout.

This approach not only aids in maintaining mathematical consistency but also simplifies solving equations and further mathematical analysis in calculus and algebra.