Mathematics Euler's Form
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Euler's Form

Euler's Form

if `theta = R` and `i = sqrt(-1)` then `e^(i theta) = cos theta + i sin theta ` is known as Euler's identity.

Now , `e^(-i theta) = cos theta - i sin theta `

Let `z=e^(i theta)`

`therefore` `|z| = 1` and `arg(z) = 0`

Also , `e^(i theta) + e^(-i theta) = 2 cos theta`

and `e^(i theta)-e^(-i theta) = 2 i sin theta`

and if , `phi in R` and `i = sqrt(-1),` then

`(i) e^(i theta) + e^(i phi) = e^((theta+phi)/2).2 cos((theta-phi)/2) `

`therefore |e^(i theta)+ e^(i phi)| = 2| cos((theta-phi)/2) |`

and `arg(e^(i theta + e^(i phi)) = ((theta+phi)/2)`

`(ii) e^(i theta)- i^(i phi) = e^(i(theta+phi)/2) . 2 i sin((theta-phi)/2)`

`therefore |e^(i theta- e^(i phi))| = 2|sin((theta-phi)/2)|`

and `arg(e^(i theta)- e^(i phi)) = (theta+ phi)/2 + pi/2` `[∵ i = e^(i n/2)]`

`text(Important Result)`

`1. e^(i theta) + 1 = e^((i theta)/2).2 cos(theta/2)`

`2. e^(i theta) - 1 = e^((i theta)/2).2i sin(theta/2)`

`3. (e^(i theta) - 1 )/(e^(i theta) + 1 ) = i tan (theta/2)`

`4. ` If `z= e e^(i theta) ; |z| = r ,` then `arg(z) = theta , barz = r e^(-itheta)`

`5.` If `|z-z_0| = 1 ,` then `z-z_0 e^(i theta)`



Applications of Euler's Form

If `x,y, theta in R` and ` i sqrt(-1) ,` then

let `z= x+ i y` `[text(cartesian form)]`

`= |z| (cos theta + i sin theta)` `[Polar form]`

`= |z|e^(i theta)` `[text(Euler's form)]`


`(i)` `text(Product of Two Complex Numbers)`

Let two complex numbers be

`z_1 = |z_1| e^(i theta_1)` and `z_2 = |z_2|e^(i theta_2) ,` where `theta_1.theta_2 in R` and `i = sqrt(-1)`

`therefore z_1.z_2 = |z_1|e^(itheta_1).|z_2|e^(itheta_2) = |z_1||z_2|e^(i(theta_1+theta_2))`

`= |z_1||z_2| (cos(theta_1+ theta2) + 2 sin(theta_1+ theta_2))`

Thus , `|z_1z_2| = |z_1||z_2|`

and `arg(z_1z_2) = theta_1 + theta_2 = arg(z_1)+ arg(z_2)`

`(ii)` `text(Division of Two Complex Numbers)`

Let two complex numbers be

`z_1 = |z_1| e^(i theta_1)` and `z_2 = |z_2|e^(i theta_2) ,` where `theta_1.theta_2 in R` and `i = sqrt(-1)`

`therefore z_1/z_2 = (|z_1|e^(i theta_1))/|z_2e^(i theta_2)| = |z_1|/|z_2| e^(i(theta_1-theta_2)) = |z_1|/|z_2| (cos(theta_1 - theta_2) + i sin(theta_1 - theta_2))`

Thus , `|z_1/z_2| = |z_1|/|z_2| , (z_2 ne 0)`

and `arg(z_1/z_2) = theta_1- theta_2 = arg(z_1) = arg(z_2)`

`(iii)` `text(Logarithm of a Complex Number)`

`log_e(z) = log_e(|z|e^(i theta)) = log_e|z| + log_e(e^(i theta))`

`= log_e|z| + i theta + log_e|z| + i arg(z)`

So, the general value of `log_e(z)`

`= log_e(z) + 2n pi i(-pi < arg z < pi)`

 
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