How to Solve Partial Fractions Expansion?

[jegues]

Homework Statement

Find the partial fractions expansion in the following form,

$$G(s) = \frac{1}{(s+1)(s^{2}+4)} = \frac{A}{s+1} + \frac{B}{s+j2} + \frac{B^{*}}{s-j2}$$

Homework Equations

The Attempt at a Solution

I expanded things out and found the following,

$$1 = A(s^{2} + 4) + B(s^{2} + (1-j2)s -j2) + B^{*}(s^{2} + (1+j2)s + j2)$$

From this I get the following equations,

$$A + B + B^{*} = 0$$

$$B(1-j2) + B^{*}(1+j2) = 0$$

$$4A - Bj2 + B^{*}j2 = 1$$

This doesn't seem like a pleasant set of equations to solve.

I did another partial fractions expansion like so,


$$G(s) = \frac{1}{(s+1)(s^{2}+4)} = \frac{D}{s+1} + \frac{Es + F}{s^{2}+4}$$

and found,

$$D = \frac{1}{5}, E = - \frac{1}{5}, F = \frac{1}{5}$$

but this doesn't give me the form I need.

Any ideas? Any convenient ways to solve this?

 

[jbunniii]

Hint: if $B^*$ is the complex conjugate of $B$, then $B+B^* = 2\text{Re}(B)$ and $B - B^* = 2i\text{Im}(B)$. You can use these facts to simplify the unpleasant equations you obtained.

 

[lurflurf]

solve by inspection

$$\frac{1}{(s+1)(s^{2}+4)} =\frac{1}{(s+1)((-1)^{2}+4)}+\frac{1}{(2j+1)(2j+2j)(s-2j)}$$
$$+\frac{1}{(-2j+1)(s+2j)(-2j-2j)}= \frac{A}{s+1} + \frac{B}{s+2j} + \frac{B^{*}}{s-2j}$$

in general

$$\prod_k \frac{1}{x-a_k}=\sum_l \frac{1}{x-a_l}\prod_{k \ne l} \frac{1}{a_l-a_k}$$

 

[SammyS]

Hint: if $B^*$ is the complex conjugate of $B$, then $B+B^* = 2\text{Re}(B)$ and $B - B^* = 2i\text{Im}(B)$. You can use these facts to simplify the unpleasant equations you obtained.

or in Engineering terms where usually $\ j=\sqrt{-1}\$, you have

$\ B - B^* = 2j\text{ Im}(B) \$

 

[jbunniii]

Sorry, I just couldn't bring myself to type $j$. :tongue:

 

[Mark44]

Sorry, I just couldn't bring myself to type $j$. :tongue:

I know what you mean. Also, when I saw j2 in the first post, I first thought he meant j². I like 2j better than j2, and 2i better than 2j.

 

[Ray Vickson]

Homework Statement

Find the partial fractions expansion in the following form,

$$G(s) = \frac{1}{(s+1)(s^{2}+4)} = \frac{A}{s+1} + \frac{B}{s+j2} + \frac{B^{*}}{s-j2}$$

Homework Equations

The Attempt at a Solution

I expanded things out and found the following,

$$1 = A(s^{2} + 4) + B(s^{2} + (1-j2)s -j2) + B^{*}(s^{2} + (1+j2)s + j2)$$

From this I get the following equations,

$$A + B + B^{*} = 0$$

$$B(1-j2) + B^{*}(1+j2) = 0$$

$$4A - Bj2 + B^{*}j2 = 1$$

This doesn't seem like a pleasant set of equations to solve.

I did another partial fractions expansion like so,


$$G(s) = \frac{1}{(s+1)(s^{2}+4)} = \frac{D}{s+1} + \frac{Es + F}{s^{2}+4}$$

and found,

$$D = \frac{1}{5}, E = - \frac{1}{5}, F = \frac{1}{5}$$

but this doesn't give me the form I need.

Any ideas? Any convenient ways to solve this?

It is probably easier to first expand as
$$\frac{1}{(s+1)(s^2+4)} = \frac{A}{s+1} + \frac{Bs+C}{s^2+4}$$
then expand
$$\frac{1}{s^2+4} = \frac{E}{s+2i} + \frac{F}{s-2i}$$
BTW: in TeX (or LaTeX) you do not need to write 's^{2}'; just plain 's^2' will do. You only need the '{.}' if the super-script (or sub-script) is more than one term. (Even a multi-letter single command can do without the '{.}'; for example, look at $s^\alpha$, which was entered as s^\alpha.)

 

[Ray Vickson]

Homework Statement

Find the partial fractions expansion in the following form,

$$G(s) = \frac{1}{(s+1)(s^{2}+4)} = \frac{A}{s+1} + \frac{B}{s+j2} + \frac{B^{*}}{s-j2}$$

Homework Equations

The Attempt at a Solution

I expanded things out and found the following,

$$1 = A(s^{2} + 4) + B(s^{2} + (1-j2)s -j2) + B^{*}(s^{2} + (1+j2)s + j2)$$

From this I get the following equations,

$$A + B + B^{*} = 0$$

$$B(1-j2) + B^{*}(1+j2) = 0$$

$$4A - Bj2 + B^{*}j2 = 1$$

This doesn't seem like a pleasant set of equations to solve.

I did another partial fractions expansion like so,


$$G(s) = \frac{1}{(s+1)(s^{2}+4)} = \frac{D}{s+1} + \frac{Es + F}{s^{2}+4}$$

and found,

$$D = \frac{1}{5}, E = - \frac{1}{5}, F = \frac{1}{5}$$

but this doesn't give me the form I need.

Any ideas? Any convenient ways to solve this?

It is probably easier to first expand as
$$\frac{1}{(s+1)(s^2+4)} = \frac{A}{s+1} + \frac{Bs+C}{s^2+4}$$
then expand
$$\frac{1}{s^2+4} = \frac{E}{s+2i} + \frac{F}{s-2i}$$
BTW: in TeX (or LaTeX) you do not need to write 's^{2}'; just plain 's^2' will do. You only need the '{.}' if the super-script (or sub-script) is more than one term. (Even a multi-letter single command can do without the '{.}'; for example, look at $s^\alpha$, which was entered as s^\alpha.