Page 42 - Aeronautical Engineer Data Book

Fundamental dimensions and units 29

3
2
1 � bx b x 2 b x 3 �
–1
(a – bx) = 33 1 + 33 + 33 + 33 + ...
a a a 2 a 3
2
2
2
(b x < a )
Exponential series
(x ln a) 2 (x ln a) 3
x
a = 1 + x ln a + 33 + 33 + ...
2! 3!
x 2 x 3
x
e = 1 + x + 33 + 33 + ...
2! 3!
Logarithmic series
1
3
2
1
ln x = (x – 1) – 33 (x – 1) + 33 (x – 1) – ... (0
2
3
< x < 2)
�
�
x – 1 � x – 1 2 � x – 1
3
ln x = 33 + 1 2 3 3 33 + 1 3 3 3 33
x x x
� 1 �
+ ... x > 33
2
�
1 x – 1
x –1 1 x –1 � � � 5
3
ln x = 2[33 . 33 33 + 33 33
x + 1 3 x + 1 5 x +1
+ ... (x positive)
x 2 x 3 x 4
ln (1 + x) = x – 33 + 33 – 33 + ...
2 3 4
Trigonometric series
x 3 x 5 x 7
sin x = x – 33 + 33 – 33 + ...
3! 5! 7!
x 2 x 4 x 6
cos x = 1 – 33 + 33 – 33 + ...
2! 4! 6!
x 3 2x 5 17x 7 62x 9
tan x = x + 33 + 33 + 33 + 33
3 15 315 2835
� 2 π 2 �
+ ... x < 33
4
1 x 3 1·3 x 5 1·3·5 x 7
sin x = x + 33 33 + 33 + 33 + 33 33
–1
2 3 2·4 5 2·4·6 7
2
+ ... (x < 1)