Thomas Calculus 13th [Solutions]

Section 10.10 The Binomial Series and Applications of Taylor Series 785
70.
i i i( )
e i e e i i
(a)
(b)
cos sin cos( ) sin( ) cos sin
i i
e e (cos i sin ) (cos i sin ) 2cos cos
e e
cosh i
2
i i
e e (cos i sin ) (cos i sin ) 2i sin i sin
e e
sinh i
2
i
i
i
i
71.
2 3 4 3 5 7
x
e sin x 1 x x x x x x x x
2! 3! 4! 3! 5! 7!
2 1 1 3 1 1 4 1 1 1 5 2 1 3 1 5
x x x x x x x x x
6 2 6 6 120 12 24 3 30
(1) (1) ;
x ix (1 i)
x x x x x
e e e e cos x i sin x e cos x i e sin x e sin x is the series of the imaginary part
of
(1 i)
x
e which we calculate next;
n
2 3 4
e (1 i)
x ( x ix) ( x ix) ( x ix) ( x ix)
1 ( )
n! x ix 2! 3! 4!
n 0
1 2 1 3 3 1 4 1 5 5 1 6
2! 3! 4! 5! 6!
1 x ix 2ix 2ix 2x 4x 4x 4ix 8ix the imaginary part of
(1 i)
x
e is
x x x x x x x x x x in agreement with our
2 2 2 3 4 5 8 6 2 1 3 1 5 1 6
2! 3! 5! 6! 3 30 90
x
product calculation. The series for e sin x converges for all values of x.
72.
d
dx
( a ib)
ax ax ax
e d e cos bx i sin bx ae cos bx i sin bx e b sin bx bi cos bx
dx
ax ax ( a ib) x ( a ib) x ( a ib)
x
ae bx i bx bie bx i bx ae ibe a ib e
cos sin cos sin ( )
i
i
73. (a) e
1e 2
cos i sin cos i sin
(b)
i
1 1 2 2
i
1 2 1 2 i 1 2 2 1 1 2 i 1 2 e
cos cos sin sin sin cos sin cos cos sin
cos i sin 1 1
cos i sin cos i sin e
e cos( ) i sin( ) cos i sin cos i sin
i
1 2
a bi ( a bi)
x
ax
e C1 iC a bi
2 e cos bx i sin bx C1 iC2
a b a b
74.
2 2 2 2
ax
e a cos bx ia sin bx ibcos bx b sin bx C
2 2 1 iC2
a b
ax
e a cos bx b sin bx a sin bx b cos bx i C
2 2 1 iC2
a b
ax
ax
e a cosbx b sin bx ie a sin bx b cos bx
C
2 2 1 iC
2 2
2 ;
a b a b
( a bi)
x ax ibx ax ax ax
e e e e cos bx i sin bx e cos bx ie sin bx , so that given
( a bi) x a bi ( a bi)
x
a b
ax
e a cos bx b sin bx
e cos bx dx C 1 and
a b
e dx e C
2 2
1 iC 2 we conclude that
2 2
ax
ax
e a sin bx b cos bx
e sin bx dx C
2 2
2
a b
ax
Copyright
2014 Pearson Education, Inc.