Perturbing a matrix with asymptotic constraints
$begingroup$
I want to expand the following function in series of $epsilon = a/l$:
$$f=frac{h cosalpha + lsinalpha}{l cos alpha + c sin alpha}$$
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 $$frac{d}{k l^3}simmathcal{O}(1)$$
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.
$$ a k/l^2.$$
matrices taylor-expansion inverse perturbation-theory
asked Jan 11 at 12:35
usumdelphiniusumdelphini
323111
$endgroup$
add a comment |
$begingroup$
I want to expand the following function in series of $epsilon = a/l$:
$$f=frac{h cosalpha + lsinalpha}{l cos alpha + c sin alpha}$$
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 $$frac{d}{k l^3}simmathcal{O}(1)$$
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.
$$ a k/l^2.$$
matrices taylor-expansion inverse perturbation-theory
asked Jan 11 at 12:35
usumdelphiniusumdelphini
323111
$endgroup$
$begingroup$
What are $a, b, c, d, l$ ?
$endgroup$
– Keith McClary
Jan 11 at 17:17
add a comment |
$begingroup$
I want to expand the following function in series of $epsilon = a/l$:
$$f=frac{h cosalpha + lsinalpha}{l cos alpha + c sin alpha}$$
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 $$frac{d}{k l^3}simmathcal{O}(1)$$
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.
$$ a k/l^2.$$
matrices taylor-expansion inverse perturbation-theory
asked Jan 11 at 12:35
usumdelphiniusumdelphini
323111
$endgroup$
I want to expand the following function in series of $epsilon = a/l$:
$$f=frac{h cosalpha + lsinalpha}{l cos alpha + c sin alpha}$$
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 $$frac{d}{k l^3}simmathcal{O}(1)$$
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.
$$ a k/l^2.$$
matrices taylor-expansion inverse perturbation-theory
matrices taylor-expansion inverse perturbation-theory
asked Jan 11 at 12:35
usumdelphiniusumdelphini
323111
asked Jan 11 at 12:35
usumdelphiniusumdelphini
323111
edited Jan 12 at 14:51
usumdelphini
asked Jan 11 at 12:35
usumdelphiniusumdelphini
323111
asked Jan 11 at 12:35
usumdelphiniusumdelphini
323111
asked Jan 11 at 12:35
usumdelphiniusumdelphini
323111
323111
$begingroup$
What are $a, b, c, d, l$ ?
$endgroup$
– Keith McClary
Jan 11 at 17:17
add a comment |
$begingroup$
What are $a, b, c, d, l$ ?
$endgroup$
– Keith McClary
Jan 11 at 17:17
$begingroup$
What are $a, b, c, d, l$ ?
$endgroup$
– Keith McClary
Jan 11 at 17:17
$begingroup$
What are $a, b, c, d, l$ ?
$endgroup$
– Keith McClary
Jan 11 at 17:17
add a comment |
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$begingroup$
What are $a, b, c, d, l$ ?
$endgroup$
– Keith McClary
Jan 11 at 17:17