# Perturbing a matrix with asymptotic constraints

I want to expand the following function in series of $$\epsilon = a/l$$:

$$f=\frac{h \cos\alpha + l\sin\alpha}{l \cos \alpha + c \sin \alpha}$$

I know from the physics of the problem that also $$\frac{b}{l},\frac{c}{l}\sim \mathcal{O}(\epsilon).$$ Finally, I want to be able to impose that $$\frac{d}{k\ l^3}\sim\mathcal{O}(1)$$

Note: $$a,b,c,l$$ are positive numbers (lengths). And $$d/k l^3$$ is dimensionless.

What substitutions would allow me to express the function $$M(a,b,c,d,l)$$ as a series expression in powers of $$\epsilon$$?

I am particularly confused about terms that do not have an explicit ratio, e.g.

$$a k/l^2.$$

• What are $a, b, c, d, l$ ? – Keith McClary Jan 11 at 17:17