Heaviside cover-up method

The Heaviside cover-up method, named after Oliver Heaviside, is one possible approach in determining the coefficients when performing the partial-fraction expansion of a rational function.

Separation of a fractional algebraic expression into partial fractions is the reverse of the process of combining fractions by converting each fraction to the lowest common denominator (LCM) and adding the numerators. This separation can be accomplished by the Heaviside cover-up method, another method for determining the coefficients of a partial fraction. Case one has fractional expressions where factors in the denominator are unique. Case two has fractional expressions where some factors may repeat as powers of a binomial.

In integral calculus we would want to write a fractional algebraic expression as the sum of its partial fractions in order to take the integral of each simple fraction separately. Once the original denominator, D₀, has been factored we set up a fraction for each factor in the denominator. We may use a subscripted D to represent the denominator of the respective partial fractions which are the factors in D₀. Letters A, B, C, D, E, and so on will represent the numerators of the respective partial fractions. When a partial fraction term has a single (i.e. unrepeated) binomial in the denominator, the numerator is called a residue.

We calculate each respective numerator by (1) taking the root of the denominator (i.e. the value of x that makes the denominator zero) and (2) then substituting this root into the original expression but ignoring the corresponding factor in the denominator. Each root for the variable is the value which would give an undefined value to the expression since we do not divide by zero.

General formula for a cubic denominator with three distinct roots:

Where x = a and

and where x = b and

and where x = c and

Factorize the expression in the denominator. Set up a partial fraction for each factor in the denominator. Apply the cover-up rule to solve for the new numerator of each partial fraction.

Set up a partial fraction for each factor in the denominator. With this framework we apply the cover-up rule to solve for A, B, and C.

1. D₁ is x + 1; set it equal to zero. This gives the residue for A when x = −1.

2. Next, substitute this value of x into the fractional expression, but without D₁.

3. Put this value down as the value of A.

Proceed similarly for B and C.

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