Geography Reference
In-Depth Information
where
1
31
517
120389
1362253
2
4
6
8
10
c
e
e
e
e
e
2
3
180
5040
181400
29937600
23
251
102287
450739
4
6
8
10
c
e
e
e
e
4
360
3780
1814400
997920
761
47561
434501
6
8
10
(49)
c
e
e
e
6
45360
1814400
14968800
6059
625511
8
10
c
e
e
8
1209600
59875200
48017
10
c
29937600
e
10
3.2. The inverse expansions using the Hermite interpolation method
In mathematical analysis, interpolation with functional values and their derivative values is
called Hermite interpolation. The processes to derive the inverse expansions using this
method are as follows:
1.
To suppose the inverse expansions are expressed in a series of the sines of the multiple
arcs with coefficients to be determined;
2.
To compute the functional values and their derivative values at specific points;
3.
To solve linear equations according to interpolation constraints and obtain the
coefficients.
The detailed derivation processes are given by Li (2008, 2010). Confined to the length of the
chapter, they are omitted. Comparing the results derived by this method with those in 3.1,
one will find that they are consistent with each other even though they are formulated in
different ways. This fact substantiates the correctness of the derived formulas.
3.3. The inverse expansions using the Lagrange's theorem method
We wish to investigate the inversion of an equation such as
y
xf x
()
(50)
with
fx
()
and
y
. The Lagrange's theorem states that in a suitable domain the
solution of (50) is
nn
1
(1)
d
n
xy
[()]
fy
(51)
n
!
n
1
dy
n
1
The proof of this theorem is given by Whittaker (1902) and Peter (2008).
The processes to derive the inverse expansions using the Lagrange series method are as
follows:
