Page 53 - Schaum's Outline of Theory and Problems of Advanced Calculus

44 FUNCTIONS, LIMITS, AND CONTINUITY [CHAP. 3

Note the analogy with real numbers, polynomials corresponding to integers, rational functions to
rational numbers, and so on.

TRANSCENDENTAL FUNCTIONS
The following are sometimes called elementary transcendental functions.

x
1. Exponential function: f ðxÞ¼ a , a 6¼ 0; 1. For properties, see Page 3.
2. Logarithmic function: f ðxÞ¼ log x, a 6¼ 0; 1. This and the exponential function are inverse
a
functions. If a ¼ e ¼ 2:71828 ... ; called the natural base of logarithms,we write
f ðxÞ¼ log x ¼ ln x, called the natural logarithm of x. For properties, see Page 4.
e
3. Trigonometric functions (Also called circular functions because of their geometric interpreta-
tion with respect to the unit circle):
sin x 1 1 1 cos x
cos x sin x cos x tan x sin x
sin x; cos x; tan x ¼ ; csc x ¼ ; sec x ¼ ; cot x ¼ ¼
The variable x is generally expressed in radians ( radians ¼ 1808). For real values of x,
sin x and cos x lie between 1and 1 inclusive.
The following are some properties of these functions:
2
2
2
2
2
2
sin x þ cos x ¼ 1 1 þ tan x ¼ sec x 1 þ cot x ¼ csc x
sinðx yÞ¼ sin x cos y cos x sin y sinð xÞ¼ sin x
cosðx yÞ¼ cos x cos y sin x sin y cosð xÞ¼ cos x
tan x tan y
tanð xÞ¼ tan x
1 tan x tan y
tanðx yÞ¼
4. Inverse trigonometric functions. The following is a list of the inverse trigonometric functions
and their principal values:
ðaÞ y ¼ sin 1 x; ð =2 @ y @ =2Þ ðdÞ y ¼ csc 1 x ¼ sin 1 1=x; ð =2 @ y @ =2Þ
ðbÞ y ¼ cos 1 x; ð0 @ y @ Þ ðeÞ y ¼ sec 1 x ¼ cos 1 1=x; ð0 @ y @ Þ
ðcÞ y ¼ tan 1 x; ð =2 < y < =2Þ ð f Þ y ¼ cot 1 x ¼ =2 tan 1 x; ð0 < y < Þ
5. Hyperbolic functions are defined in terms of exponential functions as follows. These functions
may be interpreted geometrically, much as the trigonometric functions but with respect to the
unit hyperbola.
x
e e x 1 2
2 sinh x e e
ðaÞ sinh x ¼ ðdÞ csch x ¼ ¼ x x
x
e þ e x 1 2
2 cosh x e þ e
ðbÞ cosh x ¼ ðeÞ sech x ¼ ¼ x x
x
x
sinh x e e x cosh x e þ e x
ðcÞ tanh x ¼ ¼ x x ð f Þ coth x ¼ ¼ x x
cosh x e þ e sinh x e e
The following are some properties of these functions:
2
2
2
2
2
2
cosh x sinh x ¼ 1 1 tanh x ¼ sech x coth x 1 ¼ csch x
sinhðx yÞ¼ sinh x cosh y cosh x sinh y sinhð xÞ¼ sinh x
coshðx yÞ¼ cosh x cosh y sinh x sinh y coshð xÞ¼ cosh x
tanh x tanh y
tanhð xÞ¼ tanh x
tanhðx yÞ¼
1 tanh x tanh y