Elementary Functions

 Exponential Function:

The complex exponential function is defined as:

`e^z = e^(x + iy) = e^x (\cos y + i\sin y)`

`\implies | e^z| = e^x`


Note:

  1. `e^z = e^(x + iy)` if z is real.
  2. `e^z` is analytic for all z.
  3. The derivative of `e^z` is `e^z`.
  4. `e^(z_1) . e^(z_2) = e^(z_1 + z_2)`
  5. `\frac{e^(z_1)}{e^(z_2)} = e^(z_1 - z_2)`

Euler Formula: 

`e^(iy) = (\cos y + i\sin y)`
`\implies z = r (\cos\theta + i\sin\theta)`
Note:  `e^(z + 2\pi i) = e^z . e^(2\pi i) = e^z (\cos2\pi + i\sin2\pi) = e^z . 1 = e^z`


Entire Function: 

A function 𝑓(𝑧) that is analytic for all ‘z’ is called an entire function. e.g.
𝑓(𝑧) = `e^z` is a entire function.

Referrence Book: Erwin Ereszig