Ref; https://en.wikipedia.org/wiki/Euler%27s_formula
Euler's formula is a golden identity connecting 2 rationals, 2 irrationals and a complex number.
I invite you all to comment on its usefulness and applications as well as other similar identities in mathematics.
Dear Ismat Beg ,
The complex numbers field is an extension of the real numbers field.
It means that any complex number can be represented as
z = (x,y) = x(1,0) + y(0,1), x any y are real numbers.
x is called the real part and y is called the imaginary part.
For simplicity any we assume that
(1,0) =1 and (0,1) = i, that is any complex z number is
a combination of (1,0) and (0,1)
z = (x,y) = x(1,0) + y(0,1) = x + iy.
Interesting relations in the complex field are
i×i=(0,1)×(0,1) = (-1,0) = -1(1,0) = -1
by convention, we use the notation i² = -1 or i = √(-1)
one can see
i³ = i²×i = -i, i⁴ = i²×i² =1
In general, we can prove:
i^{4k} =1, i^{4k+1} = i, i^{4k+2} = -1, i^{4k+3} = -i.
To study functions in complex field we need to extend the notions
of real functions in real field into the complex field,
as sinx, cosx, tanx, ln(x),Log(x), sinh(x),cosh(x), polynomials, rational functions and the exponential function ex etc.
In the real field, using the Maclaurin expansion series,
we can expand ex=1+x+((x²)/(2!))+..+((xⁿ)/(n!)) +..
cos(x) = 1 - (x²)/(2!) + (x⁴)/(4!) - ..
sin(x) = x - (x³)/(3!)+ (x⁵)/(5!)+..
Euler's observed that
eix = 1 + (ix)+(ix)²/(2!) + (ix)³ /(3!) +(ix)⁴/(4!)..+(ix)ⁿ/(n!) +..
= 1 + ix - (x)²/(2!) - i(x³)/(3!) + (x⁴)/(4!)..+ (ix)ⁿ/(n!) +..
= (1 - (x²)/(2!)+ (x⁴)/(4!)..) + i(x - (x³)/(3!) + (x⁵)/(5!)..) = cos(x) + isin x
This formula has important implications:
(1) It provides a strong tool to describe and
simplify relations in the complex field.
for example, we describe the roots of Zⁿ = z₀ completely,
and we define all complex trigonometric functions.
(2) In many applications, as in control theory and dynamical systems, economic optimization problems, etc,
we need to study the real or complex roots of the characteristic polynomials ( called eigenvalues) this obligated us to use this important identity eix = cos(x) + isin x
The literature is full of many applications in this concern.
Best regards
Dear Ismat Beg ,
The complex numbers field is an extension of the real numbers field.
It means that any complex number can be represented as
z = (x,y) = x(1,0) + y(0,1), x any y are real numbers.
x is called the real part and y is called the imaginary part.
For simplicity any we assume that
(1,0) =1 and (0,1) = i, that is any complex z number is
a combination of (1,0) and (0,1)
z = (x,y) = x(1,0) + y(0,1) = x + iy.
Interesting relations in the complex field are
i×i=(0,1)×(0,1) = (-1,0) = -1(1,0) = -1
by convention, we use the notation i² = -1 or i = √(-1)
one can see
i³ = i²×i = -i, i⁴ = i²×i² =1
In general, we can prove:
i^{4k} =1, i^{4k+1} = i, i^{4k+2} = -1, i^{4k+3} = -i.
To study functions in complex field we need to extend the notions
of real functions in real field into the complex field,
as sinx, cosx, tanx, ln(x),Log(x), sinh(x),cosh(x), polynomials, rational functions and the exponential function ex etc.
In the real field, using the Maclaurin expansion series,
we can expand ex=1+x+((x²)/(2!))+..+((xⁿ)/(n!)) +..
cos(x) = 1 - (x²)/(2!) + (x⁴)/(4!) - ..
sin(x) = x - (x³)/(3!)+ (x⁵)/(5!)+..
Euler's observed that
eix = 1 + (ix)+(ix)²/(2!) + (ix)³ /(3!) +(ix)⁴/(4!)..+(ix)ⁿ/(n!) +..
= 1 + ix - (x)²/(2!) - i(x³)/(3!) + (x⁴)/(4!)..+ (ix)ⁿ/(n!) +..
= (1 - (x²)/(2!)+ (x⁴)/(4!)..) + i(x - (x³)/(3!) + (x⁵)/(5!)..) = cos(x) + isin x
This formula has important implications:
(1) It provides a strong tool to describe and
simplify relations in the complex field.
for example, we describe the roots of Zⁿ = z₀ completely,
and we define all complex trigonometric functions.
(2) In many applications, as in control theory and dynamical systems, economic optimization problems, etc,
we need to study the real or complex roots of the characteristic polynomials ( called eigenvalues) this obligated us to use this important identity eix = cos(x) + isin x
The literature is full of many applications in this concern.
Best regards
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