Partial fractions

Partial Fractions

Understanding Partial Fractions

  • Partial fractions are the expression of a fraction as the sum or difference of simpler fractions.
  • Any proper rational function (where the degree of the numerator is less than the degree of the denominator) can be expressed as a sum of simpler fractions.
  • The process of converting a rational function into an equivalent sum of simpler fractions is called partial fraction decomposition.

Breakdown of Rational Expressions

  • For simple linear factors in the denominator, the corresponding partial fraction has the form A/(ax+b).
  • For repeated linear factors, each factor corresponds to a separate term in the partial fraction decomposition, with powers ranging from 1 up to the multiplicity of the root.
  • When the denominator has irreducible quadratic factors, the corresponding partial fraction has the form (Ax+B)/(ax^2+bx+c).

Steps to Find the Partial Fraction Decomposition

  • Start by factoring the denominator of the rational function fully. This identifies the individual fractions to which the rational function will be decomposed.
  • Each factor, or each different factor raised to any power, contributes to a term in the partial fraction decomposition.
  • Equate the rational function to the sum of its partial fractions, and collect like terms.
  • Equating coefficients for the corresponding powers of x gives a system of linear equations in the coefficients of the partial fractions.
  • Solve the system of linear equations to find the coefficients of the partial fractions.

Completing the Square in Denominator

  • When the denominator has a quadratic factor that can’t be easily factored, use completing the square to turn it into a form where you can recognise a standard integral to be able to integrate.

Role of Partial Fractions in Calculus

  • Integration and differential equations often become much simpler when the integrand is expressed as partial fractions rather than as a single, more complex fraction.
  • Notably, partial fractions are used in the integration of rational functions and in finding inverse Laplace transforms.

Vital Reminder

  • It’s crucial to remember that a partial fraction decomposition always exists for a proper rational function. However, such a decomposition may not exist if the degree of the numerator is greater than or equal to the degree of the denominator. This is a common mistake to avoid.