Higher Engineering Mathematics

HYPERBOLIC FUNCTIONS 49

A

Let x=1,

then ch 1= 1 +

12

2 × 1

+

14

4 × 3 × 2 × 1

+

16

6 × 5 × 4 × 3 × 2 × 1

+ ···

= 1 + 0. 5 + 0. 04167 + 0. 001389 + ···

i.e. ch 1= 1. 5431 , correct to 4 decimal places,
which may be checked by using a calculator.

Problem 16. Determine, correct to 3 deci-

mal places, the value of sh 3 using the series

expansion for shx.

shx=x+

x^3

3!

+

x^5

5!

+···from above

Letx=3, then

sh 3= 3 +

33

3!

+

35

5!

+

37

7!

+

39

9!

+

311

11!

+···

= 3 + 4. 5 + 2. 025 + 0. 43393 + 0. 05424

+ 0. 00444 +···

i.e. sh 3= 10. 018 , correct to 3 decimal places.

Problem 17. Determine the power series for

2ch

(

θ

2

)

−sh 2θas far as the term inθ^5.

In the series expansion for chx, letx=

θ

2

then:

2ch

(

θ

2

)

= 2

[

1 +

(θ/2)^2

2!

+

(θ/2)^4

4!

+···

]

= 2 +

θ^2

4

+

θ^4

192

+···

In the series expansion for shx, letx= 2 θ, then:

sh 2θ= 2 θ+

(2θ)^3

3!

+

(2θ)^5

5!

+···

= 2 θ+

4

3

θ^3 +

4

15

θ^5 +···

Hence

ch

(

θ

2

)

−sh 2θ=

(

2 +

θ^2

4

+

θ^4

192

+···

)

−

(

2 θ+

4

3

θ^3 +

4

15

θ^5 +···

)

= 2 − 2 θ+

θ^2

4

−

4

3

θ^3 +

θ^4

192

−

4

15

θ^5 +···as far the

term inθ^5

Now try the following exercise.

Exercise 27 Further problems on series

expansions for coshxand sinhx

1. Use the series expansion for chxto evalu-
   ate, correct to 4 decimal places: (a) ch 1. 5
   (b) ch 0. 8 [(a) 2.3524 (b) 1.3374]


2. Use the series expansion for shxto evaluate,
   correct to 4 decimal places: (a) sh 0.5 (b) sh 2

[(a) 0.5211 (b) 3.6269]

3. Expand the following as a power series as far
   as the term inx^5 : (a) sh 3x(b) ch 2x
   ⎡

⎢

⎣

(a) 3x+

9

2

x^3 +

81

40

x^5

(b) 1+ 2 x^2 +

2

3

x^4

⎤

⎥

⎦

In Problems 4 and 5, prove the given identities,

the series being taken as far as the term inθ^5 only.

4. sh 2θ−shθ≡θ+

7

6

θ^3 +

31

120

θ^5

5. 2 sh

θ

2

−ch

θ

2

≡− 1 +θ−

θ^2

8

+

θ^3

24

−

θ^4

384

+

θ^5

1920