An exponentiation is relating two numbers, the base a and the exponent n. The n value is a positive integer and exponentiation corresponds to frequent multiplication or a product of n factors.The exponent is written as superscript to the right of the base to the nth power. The power p is defined as when n is negative integer for non-zero. When the base a is positive real number, p is defined for exponential function.
Formulas for natural exponential function
ex = `(x)/(1!)` + `(x^(2))/(2!) + .................`
e0 = 1
d(ex) = ex
In trigonometry function the natural exponential function,
cos (x) =`(e^(ix) + e^(-ix))/(2)`
sin (x) = `(e^(ix) - e^(-ix))/(2i)`
In complex number the natural exponential function:
The symbol eiθ or exp (iθ) (called exponential of iθ) is defined by
eiθ = cos θ + i sin θ
This relation is known as Euler’s formula.
If z ≠ 0 then z = r (cos θ + i sin θ) = reiθ. This is called the exponential
form of the complex number z. By straight forward multiplication of
eiθ1 = (cos θ1 + i sin θ1) and eiθ2 = cos θ2 + i sin θ2
we have eiθ1.eiθ2 = ei(θ1+ θ2)
Example problems for natural exponential function
Problem for natural exponnential function 1:
Expand the term e2x in natural exponential function.
Solution:
The given function is e2x we have to expand the term using the formula
Solution:
ex = `(x)/(1!)` + `(x^(2))/(2!) + ....................`
e2x = 1 + `(2x)/(1!)` + `(2x^(2))/(2!) + ..............................`
= 1 + `(2x)/(1!)` + `4x^(2)/(2!) + .......................`
= 1 + 2x + 2 x2 + ..................
Problem for natural exponnential function 2:
Expand the term e4x in natural exponential function.
Solution:
The given function is e4x we have to expand the term using the formula
ex = `(x)/(1!)` + `(x^(2))/(2!) + ...................`
e4x = 1 + `(4x)/(1!)` + `(4x^(2))/(2!) + .........................`
= 1 + `(4x)/(1!)` + `16x^(2)/(2!) + ............................`
= 1 + 4x + 8 x2 + ..................
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