Rational fractions

Learning tasks

A) The field of rational fractions \(\mathbb{K}(X)\)
1) Reducing a rational fraction to irreducible form	Ex 1 Ex 2 Ex 3
2) Computing in \(\mathbb{K}(X)\)	Ex 4 Ex 5 Ex 6
B) Degree\(\virgule\) partie entière\(\virgule\) zeros and poles
3) Computing the partie entière	Ex 7 Ex 8 Ex 9
4) Identifying zeros and poles	Ex 10 Ex 11 Ex 12 Ex 13
5) Computing the degree	Ex 14 Ex 15 Ex 16
C) Partial-fraction decomposition over \(\mathbb{C}\)
6) Decomposing with simple poles	Ex 17 Ex 18 Ex 19 Ex 20
7) Decomposing with a double pole	Ex 21 Ex 22 Ex 23
8) Using the \(P' / P\) formula	Ex 24 Ex 25 Ex 26
D) Partial-fraction decomposition over \(\mathbb{R}\)
9) Decomposing with a quadratic factor	Ex 27 Ex 28 Ex 29 Ex 30
10) Bridging \(\mathbb{C}\) to \(\mathbb{R}\) by grouping conjugate poles	Ex 31 Ex 32 Ex 33 Ex 34
11) Using parity	Parity rules --- summary For a real rational fraction \(R\) with paired real simple poles at \(\pm \lambda\) (with \(\lambda \neq 0\)), write the partial-fraction terms as \(\frac{c}{X - \lambda} + \frac{d}{X + \lambda}\). Then: If \(R\) is even: \(d = -c\). If \(R\) is odd: \(d = c\). For an irreducible quadratic factor of the even form \(X^2 + q\) (with \(q > 0\)), the associated 2nd-species term \(\frac{\alpha X + \beta}{X^2 + q}\) inherits the parity of \(R\): If \(R\) is even: \(\alpha = 0\) (no \(X\) term in the numerator). If \(R\) is odd: \(\beta = 0\) (no constant term in the numerator). For two distinct irreducible quadratic factors \(Q_1, Q_2\) that are exchanged by \(X \mapsto -X\) (typical case: \(X^2 + p X + q\) and \(X^2 - p X + q\)), the corresponding 2nd-species numerators \(\alpha_1 X + \beta_1\) and \(\alpha_2 X + \beta_2\) are linked by: If \(R\) is even: \(\alpha_2 = -\alpha_1\) and \(\beta_2 = \beta_1\). If \(R\) is odd: \(\alpha_2 = \alpha_1\) and \(\beta_2 = -\beta_1\). Ex 35 Ex 36 Ex 37
E) Primitives and \(k\)-th derivatives of rational functions
12) Computing primitives via decomposition	Ex 38 Ex 39 Ex 40 Ex 41
13) Computing \(k\)-th derivatives	Ex 42 Ex 43 Ex 44

Lesson
Text book

Exercises Correction

Conventions

Throughout this exercise sheet, unless otherwise stated, \(\mathbb{K}\) denotes either \(\mathbb{R}\) or \(\mathbb{C}\). The notation \(A \wedge B\) stands for the gcd of two polynomials (chapter Arithmetic of polynomials). For a non-zero rational fraction \(R = A / B\) in irreducible form, the degree is \(\deg R = \deg A - \deg B\), and zeros and poles are always read after reducing \(R\) to irreducible form. We use both the English term polynomial part and its French equivalent partie entière for the unique polynomial \(E\) such that \(R - E\) has negative degree.

A) The field of rational fractions \(\mathbb{K}(X)\)
    1) Reducing a rational fraction to irreducible form Ex 1 Ex 2 Ex 3
    2) Computing in \(\mathbb{K}(X)\) Ex 4 Ex 5 Ex 6
B) Degree\(\virgule\) partie entière\(\virgule\) zeros and poles
    3) Computing the partie entière Ex 7 Ex 8 Ex 9
    4) Identifying zeros and poles Ex 10 Ex 11 Ex 12 Ex 13
    5) Computing the degree Ex 14 Ex 15 Ex 16
C) Partial-fraction decomposition over \(\mathbb{C}\)
    6) Decomposing with simple poles Ex 17 Ex 18 Ex 19 Ex 20
    7) Decomposing with a double pole Ex 21 Ex 22 Ex 23
    8) Using the \(P' / P\) formula Ex 24 Ex 25 Ex 26
D) Partial-fraction decomposition over \(\mathbb{R}\)
    9) Decomposing with a quadratic factor Ex 27 Ex 28 Ex 29 Ex 30
    10) Bridging \(\mathbb{C}\) to \(\mathbb{R}\) by grouping conjugate poles Ex 31 Ex 32 Ex 33 Ex 34
    11) Using parity

Parity rules --- summary

For a real rational fraction \(R\) with paired real simple poles at \(\pm \lambda\) (with \(\lambda \neq 0\)), write the partial-fraction terms as \(\frac{c}{X - \lambda} + \frac{d}{X + \lambda}\). Then:

If \(R\) is even: \(d = -c\).
If \(R\) is odd: \(d = c\).
For an irreducible quadratic factor of the even form \(X^2 + q\) (with \(q > 0\)), the associated 2nd-species term \(\frac{\alpha X + \beta}{X^2 + q}\) inherits the parity of \(R\):

If \(R\) is even: \(\alpha = 0\) (no \(X\) term in the numerator).
If \(R\) is odd: \(\beta = 0\) (no constant term in the numerator).
For two distinct irreducible quadratic factors \(Q_1, Q_2\) that are exchanged by \(X \mapsto -X\) (typical case: \(X^2 + p X + q\) and \(X^2 - p X + q\)), the corresponding 2nd-species numerators \(\alpha_1 X + \beta_1\) and \(\alpha_2 X + \beta_2\) are linked by:

If \(R\) is even: \(\alpha_2 = -\alpha_1\) and \(\beta_2 = \beta_1\).
If \(R\) is odd: \(\alpha_2 = \alpha_1\) and \(\beta_2 = -\beta_1\).

Ex 35 Ex 36 Ex 37
E) Primitives and \(k\)-th derivatives of rational functions
    12) Computing primitives via decomposition Ex 38 Ex 39 Ex 40 Ex 41
    13) Computing \(k\)-th derivatives Ex 42 Ex 43 Ex 44

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