It can be expanded into a converging infinite power series.
The binomial expansion of (1 – x)q for |x| < 1
is [HMS, eq. 5.28] [GR7, eq. 1.110]:
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That is (1 + x) raised to the q (lower case Q) power.
Note that (q–1) only happens when k > 1,
(q–2) only happens when k > 2,
etc.
Setting x = –e²sin²B and q = –3/2 gives [YST, eq. 3.5.14]:
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The double factorial (!!) multiplies every other number,
not every number. For example, 7!! = 7 × 5 × 3 × 1 = 105.
The sine function is a series, and
de Moivre's formula [JDH, 2.1.1.2.3]
[GR7, 1.316.1]
[PM, 2.2.3-14]
is used to define a series that replaces sine functions
which have even powers [GR7, 1.320] [JDH, 2.4.1.7.5]
[PM, 2.2.3-6].
That series replaces the multi-power sine function with terms
that include single-power cosines and the binomial theorem.
The binomial theorem is covered in
[GR7, p. xliii] [JDH, § 1.2.1] [PM, p. 10].
Single-power cosines therefore replace multi-power sines.
Integration then converts the single-power cosines to
single-power sines.
The [GKMP] formulas are similar, but calculate meridional arc length to
the pole instead of to the equator.
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[GKMP]
|
Grafarend, E.W., and Krumm, F.W., 2006,
Map Projections: Cartographic Information Systems,
Springer.
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[YST]
|
Yang, Q., Snyder, J.P., and Tobler, W.R., 2000,
Map Projection Transformation: Principles and Applications,
Taylor & Francis.
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[BGS]
|
Bugayevskiy, L.M., and Snyder, J.P., 1995,
Map Projections – A Reference Manual,
Taylor & Francis.
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[SMP]
|
Snyder, J.P.,
Map Projections – A Working Manual,
US Geological Survey Professional Paper 1395.
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Binomial series expansion:
|
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[GR7]
|
Gradshteyn, I.S. and Ryzhik, I.M., 2007,
Table of Integrals, Series, and Products
7th ed. (Jeffrey and Zwillinger, eds.),
p. 25, eq. 1.110.
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[JDH]
|
Jeffrey, A., and Dai, H.H., 2008,
Handbook of Mathematical Formulas
and Integrals 4th ed.,
p. 75, eq. 1.8.3.2.1.
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[HMS]
|
Hassani, S., 2000,
Mathematical Methods for Students of
Physics and Related Fields,
p. 232, eq. 5.28.
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[ABS]
|
Abramowitz, M., and Stegun, I.A., 1972,
Handbook of Mathematical Functions,
p. 15, eq. 3.6.9.
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[PM]
|
Polyanin, A.D., and Manzhirov, A.V., 2007,
Handbook of Mathematics for Engineers and Scientists,
p. 10.
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[SPL]
|
Spiegel, M.R., and Liu, J., 1999,
Mathematical Handbook of Formulas and Tables
2nd ed.,
p. 135, eq. 22.4.
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[BH5]
|
Bird, J., 2006,
Higher Engineering Mathematics
5th ed.,
§ 7.2.
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Trigonometry, de Moivre:
|
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[FCV]
|
Fisher, S.D., 1999,
Complex Variables
2nd ed.,
pp. 6-7.
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[MQ]
|
McQuarrie, D.A., 2003,
Mathematical Methods for Scientists and Engineers,
p. 173.
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[BGW]
|
Bali, N., Goyal, C., and Watkins, C., 2007,
Advanced Engineering Mathematics
7th ed.,
§ 1.9.
|
Powers of trigonometric functions:
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[GR7]
|
Gradshteyn and Ryzhik,
p. 31, eq. 1.320.1.
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[JDH]
|
Jeffrey and Dai,
§§ 2.1.1.3,
2.4.1.6,
2.4.1.7.
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[PM]
|
Polyanin and Manzhirov,
p. 28.
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