Euler's Formula and converting complex numbers

[Relz]

I have an independent study unit about complex numbers and I have a few questions that I don't understand. They are probably simple once I understand but for now:

Express 3i in exponential form

I'm not sure what to do when there isn't a real number attached and vice versa.

 

[Idealistic]

I have an independent study unit about complex numbers and I have a few questions that I don't understand. They are probably simple once I understand but for now:

Express 3i in exponential form

I'm not sure what to do when there isn't a real number attached and vice versa.

any complex numbers z = x + iy can be expressed in the form:

e^z = e^x+iy or equivalently,

e^x(cos(y) + isin(y)) *

In words: e to the power of the "real" compnonent, x, multplied by the sum of cos(imaginary part) and i times the sin(imaginary part).

To verify this, take the derivative of *

Note: if x = 0, then e⁰ = 1 and cos(y) + isin(y) = e^iy

 

[Relz]

any complex numbers z = x + iy can be expressed in the form:

e^z = e^x+iy or equivalently,

e^x(cos(y) + isin(y)) *

In words: e to the power of the "real" compnonent, x, multplied by the sum of cos(imaginary part) and i times the sin(imaginary part).

To verify this, take the derivative of *

Note: if x = 0, then e⁰ = 1 and cos(y) + isin(y) = e^iy

Okay thank you!