In my proof that the complicated formula for nλ reduces to s sinθ I chose the big-O notation over the Taylor expansion. Halfway through I realized that I was going to unnecessary high order because S=R+O-ε to some power. That lack of forethought caused me to include terms which would only confuse the reader. Here I place what I think is the extra terms that were deleted from the essay:
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In order to ensure that the first order calculation is sufficient in the MyOpenMath version of this question, the software verifies that the first order solution is within 10% of the true answer. It is convenient to define
so that:
Note from the figure that
, and that the two paths are effectively parallel when
. The exact formula for the path difference is:
where,
Apparently, if the two paths are nearly parallel, we should be able to show that:
.
To verify this we can perform a Taylor series of
or equivalently use this expansion for small
:
The first four terms on the RHS refer to the zeroth, first, second, and third order terms, respectively. The last (fifth-order) term will be briefly discussed but not calculated.
Replacing
by
into the aforementioned expression, we see that the zeroth, second, and fourth order terms cancel when we subtract:
![{\displaystyle (1+\epsilon )^{1/2}-(1-\epsilon )^{1/2}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf5f50a8a0721a2f6a91e1ee0054b1ba85fb39c)
![{\displaystyle 1+{\frac {\epsilon }{2}}-{\frac {\epsilon ^{2}}{8}}+{\frac {\epsilon ^{3}}{16}}+{\mathcal {O}}(\epsilon ^{4})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9916bde1bded4e0711490a6d141287c78653dfbc)
I replaced
by
because it is obvious that the fourth order terms also cancel due to the subtraction. In order to obtain a useful value for our small parameter
, we divide our expression for
by
:
where we have defined the small parameter,