Perturbing a matrix with asymptotic constraints












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$begingroup$


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.



$$ a k/l^2.$$










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
















0












$begingroup$


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.



$$ a k/l^2.$$










share|cite|improve this question











$endgroup$












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














0












0








0





$begingroup$


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.



$$ a k/l^2.$$










share|cite|improve this question











$endgroup$




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






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share|cite|improve this question













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edited Jan 12 at 14:51







usumdelphini

















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


















  • $begingroup$
    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










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