EXPONENTIAL AND LOGARITHM SERIES
Exponential Function : A function ex is called an exponential function. It can be expanded with different powers of e. Some results of exponential series are given below :
x2 x3
(a) ex = l + x +—- + —- + …¥
2! 3!
x2 x3
(b) e-x = l – x + —- – —- + …¥
2! 3!
ex + e-x x2 x4
(c) ——— = 1 + —- + —- + …¥ 2 2! 3!
ex – e-x x3 x5
(d) ——— = x + —- + —- + …¥ 2 3! 5!
Some modified results of exponential functions-
1 1 1 n ¥ 1
(1) e1 = 1 + 1 + —- + —- + …¥ = Lim
n®¥
1 + —
= å —
21 3! n n=0 n
1 1 1 1 n ¥ 1
(2) e-1 = —- + —- + —- + …¥ = Lim
n®¥
1 – —
= å (-1)n —
2! 3! 4! n n=0 n
e1 – e-1 1 1
(3) ———- = 1 + —- + —- + …¥ 2 3! 4!
e1 – e-1 1 1
(4) ———- = 1 + —- + —- + …¥ 2 3! 5!
Some conclusions of exponential functions :
- Value of e lies 2 < e < 3
- e is an irrational number
- If loge a = m then a = e and a = e
m x mx
mx m2x2 m3x3
= 1 + —— + —— + —— +…¥ 1! 2! 3!
x x2
or ax = 1 + —– (log a) + —– (log a)2 +…
2 c
1! 2!
- eix = cosx + sinx
(c) e-ix = cosx – isin x
LOGARITHMIC SERIES
Naperian or natural logarithmic are the logarithmis of numbers calculated to the base e. Logarithmis to the base 10 are known as common logarithm.
|
The Naperian logarithms can be converted to common logarithms as follows log n 1
|
logn
= ——– = 0.43429448. (logen) because———- = 0.43429448
log 10 log 10
e e
This relation shows that the log of any number to the base 10 can be obtained by multiplying the log of the number to the base e by 0.43429448
Some logarithims expansions
- loge (1 + x) = x2 x3 x4
x – —– + —– – —– + … ¥
2 3 4
- loge (1 – x) = x2 x3 x4
-x – – – …¥
2 3 4
- loge (1 + x) – loge (1 – x)
1 + x x3 x5
= log ——– = 2 x + —- + —- + … ¥
1 – x 3 5
1
- loge 1 +—— =
n
1 1 1
2 + + + … ¥
2n + 1 3(2n + 1)3 5(2n + 1)5
- loge (1 + x) loge (1 – x) = loge (1 – x )
2
x2 x4
= -2 —– + —- + …. ¥
2 4
1 1 1 1
(f) log2 = 1 – —- + —- – —- + —- – …
2 3 4 5
