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{sin(x) uxx + 2 cos(x) ux }
uyy =
L2

— sin4 (x)/L3
uyyy

ux uxx uxxx

8 sin2 ’6 ’6 cos sin ’ sin2
— sin5 (x)/L4
uyyyy

ux uxx uxxx uxxxx

24 cos ’48 cos sin2 36 sin ’44 sin3 12 cos sin2 sin3

— sin6 (x)/L5
uyyyyy

ux uxx uxxx uxxxx uxxxxx

’384 sin4 ’20 cos sin3 ’ sin4
400 cos sin3 140 sin4
+480 sin2 ’120 ’240 cos sin ’120 sin2
[— sin7 /L6 ]
uyyyyyy

ux uxx uxxx uxxxx uxxxxx uxxxxxx

3840 cos sin4 1200 cos sin2 300 sin3 30 cos sin4 sin5
1800 sin
’3840 cos sin2 ’6000 sin3 ’1800 cos sin4 ’340 sin5
+4384 sin5
+720 cos


[— sinq /Lp ] denotes that all entries in the box below must be multiplied by this factor.
APPENDIX E. CHANGE-OF-COORDINATE DERIVATIVE TRANSFORMATIONS
556





Table E.5: Transformations of derivatives for the mapping y = L x/ 1 ’ x2 where L is a
constant map parameter. This transformation converts a series of T Bn (y) into a Chebyshev
series in x, that is, T Bn (y) = Tn (x), ∀n, y ∈ [’∞, ∞], x ∈ [’1, 1]. De¬ning the auxiliary
parameter Q(x) ≡ 1 ’ x2 greatly simpli¬es the tables.


uy = Q Q ux /L


uyy = Q2 {Q uxx ’ 3 x ux } /L2


Q Q2 Q2 uxxx ’ 9 x Q uxx + (12 ’ 15 Q) ux /L3
uyyy =


uyyyy = Q3 Q3 uxxxx ’ 18 x Q2 uxxx + (75 Q ’ 87 Q2 ) uxx
+(105 x Q ’ 60 x) ux } /L4


Q Q3 Q4 uxxxxx ’ 30 x Q3 uxxxx + (255 Q2 ’ 285 Q3 ) uxxx
uyyyyy =
+x Q(975 Q ’ 660) uxx + (360 ’ 1260 Q + 945 Q2 ) ux /L5


uyyyyyy = Q4 Q5 uxxxxxx ’ 45 x Q4 uxxxxx + Q3 (645 ’ 705 Q) uxxxx
+x Q2 (4680 Q ’ 3465) uxxx + Q (6300 ’ 18585 Q + 12645 Q2 ) uxx
’x (2520 ’ 11340 Q + 10395 Q2 ) ux /L6
557




Table E.6: Transformations of derivatives for the mapping y = L cot2 (x/2) which converts
a rational-Chebyshev series in T Ln (y) into a Fourier cosine series in cos(nx). L is a
constant, the “map parameter”. Note that all sines and cosines in the table have arguments
of (x/2), not x.

Note: the [— sinq /{2n Lp cosr }] denotes that all entries in the box below must be multiplied
by this common factor.

uy = ’{sin3 (x/2)/[L cos(x/2)]} ux


sin5 (x/2) x x x
uxx + 3 ’ 2 sin2
uyy = 2 cos sin ux
2 L2 cos3 (x/2) 2 2 2



— sin7 (x/2)/{4 L3 cos5 (x/2)}
uyyy

ux uxx uxxx

’8 sin4 +20 sin2 ’15 12 cos sin3 ’18 cos sin 4 sin4 ’4 sin2


— sin9 /{8 L4 cos7 }
uyyyy

ux uxx uxxx uxxxx

’48 sin6 +168 sin4 88 cos sin5 ’232 cos sin3 48 sin6 ’120 sin4 ’8 cos sin5
’210 sin2 +105 +72 sin2 +8 cos sin3
+174 cos sin


— sin11 /{16 L5 cos9 }
uyyyyy

ux uxx uxxx uxxxx uxxxxx

’384 sin8 +1728 sin6 ’160 cos sin7 ’16 sin8
800 cos sin7 560 sin8
’3024 sin4 ’2960 cos sin5 ’2080 sin6 +400 cos sin5 +32 sin6
+2520 sin2 ’945 ’240 cos sin3 ’16 sin4
+3900 cos sin3 +2660 sin4
’1950 cos sin ’1140 sin2
APPENDIX E. CHANGE-OF-COORDINATE DERIVATIVE TRANSFORMATIONS
558




Table E.7: Transformations of derivatives for the mapping y = L(1 + x)/(1 ’ x) + r0 where
y ∈ [r0 , ∞] and x ∈ [’1, 1]. This is similar to the map of Table E.6 except that it takes
the semi-in¬nite interval in y to the standard interval for Chebyshev polynomials (rather
than that for Fourier series). De¬ning the auxiliary parameter Q(x) ≡ x2 ’ 2x + 1 greatly
simpli¬es the tables. Note that the translation of the origin given by r0 disappears when
the mapping is differentiated, and therefore all entries below are independent of r0 .


Q
u y = ux
2L

Q
uyy = {Q uxx + 2 (x ’ 1) ux }
4 L2

Q2
= {Q uxxx + 6 (x ’ 1) uxx + 6 ux }
uyyy
8 L3

Q2
uyyyy = Q uxxxx + 12 (x ’ 1) Q uxxx + 36 Q uxx + 24 (x ’ 1) ux
2
16 L4


uyyyyy = Q2 uxxxxx + 20 (x’1) Q uxxxx + 120 Q uxxx + 240 (x’1) uxx
Q3
+120 ux }
32 L5


uyyyyyy = Q3 uxxxxxx + 30 (x ’ 1) Q2 uxxxxx + 300 Q2 uxxxx
Q3
+1200 (x ’ 1) Q uxxx + 1800 Q uxx + 720 (x ’ 1) ux }
64 L6
559




Table E.8: Transformation of derivatives for the mapping y = L arctanh(x) which converts
polynomials in sech(y/L) and tanh(y/L) into ordinary polynomials in x. In particular,
tanh(y/L) = x & sech2 (y/L) = 1 ’ x2 The auxiliary parameter is Q(x) ≡ 1 ’ x2


Q
uy = u x
L

Q
uyy = {Q uxx ’ 2 x ux }
L2

Q
uyyy = Q2 uxxx ’ 6 x Q uxx + (4 ’ 6 Q) ux
L3

Q
uyyyy = Q3 uxxxx ’ 12 x Q2 uxxx + Q (28 ’ 36 Q) uxx + x (24 Q ’ 8) ux
L4


u5y = Q4 uxxxxx ’ 20 x Q3 uxxxx + Q2 (100 ’ 120 Q) uxxx
Q
+x Q (240 Q ’ 120) uxx + (16 ’ 120 Q + 120 Q2 ) ux
L5


u6y = Q5 uxxxxxx ’ 30 x Q4 uxxxxx + Q3 (260 ’ 300 Q) uxxxx
+x Q2 (1200 Q ’ 720) uxxx + Q (496 ’ 2160 Q + 1800 Q2 ) uxx
Q
’x (32 ’ 480 Q + 720 Q2 ) ux
L6
APPENDIX E. CHANGE-OF-COORDINATE DERIVATIVE TRANSFORMATIONS
560




Table E.9: Transformation of derivatives under a general mapping x = f (r), y = g(s)


J ≡ fr gs ’ fs gr (E.6)


gs ur ’ gr us ’fs ur + fr us
‚u ‚u
(E.7)
= , =
‚x J ‚y J

gs gs urr ’ 2gr gs urs + gr gr uss
uxx =
J2
gs gs grr ’ 2gr gs grs + gr gr gss
’uy
J2
gs gs frr ’ 2gr gs fss + gr gr fss
’ux (E.8)
J2


fs fs urr ’ 2fr fs urs + fr fr — uss
uyy =
J2
fs fs grr ’ 2fr fs grs + fr fr gss
’uy
J2
fs fs frr ’ 2fr fs fss + fr fr fss
’ux (E.9)
J2


Jr = frr gs + fr grs ’ frs gr ’ fs grr ; , Js = frs gs + fr gss ’ fss gr ’ fs grs ; (E.10)



((fr gs + fs gr )urs ’ fr gr uss ’ fs gs urr )
uxy =
J2
(fr gss ’ fs grs ) (fs gs Jr ’ fr gs Js )
+ + ur
J2 J3
(fs grr ’ fr grs ) (fr gr Js ’ fs gr Jr )
(E.11)
+ + us
J2 J3
Appendix F

Cardinal Functions

[of the Whittaker cardinal function, sinc(x)]: “a function of royal blood . . . whose distin-
guished properties separate it from its bourgeois brethren.”
” Sir Edmund Whittaker (1915)



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