Why do we use Complex Wave function?

[SiuKing666]

Given that we can satisfy the wave equation with a simple sine & cosine wave function (the real part of the complex wave function) in classical mechanics, why do we use the complex wave function in EM theories?
In QM it is obvious that it gets more mathematically more consistent.
Out of curiosity.

 

[DuckAmuck]

Is this what you're talking about?
https://en.wikipedia.org/wiki/Plane_wave#Complex_exponential_form

 

[lychette]

Given that we can satisfy the wave equation with a simple sine & cosine wave function (the real part of the complex wave function) in classical mechanics, why do we use the complex wave function in EM theories?
In QM it is obvious that it gets more mathematically more consistent.
Out of curiosity.

Once you are familiar with complex notation you will ask...'why do we use sine and co sine' Feynman lectures in physics are a perfect illustration of the power of complex algebra in physics...look at them if you get the chance.

 

[Philip Wood]

I'd say that a major convenience is easy factorisation. For example $$e^{i(\omega t - kx)} = (e^{i \omega t})(e^{-ikx})$$ It's all much messier using sines and cosines. An insight I found useful – I know this is handwaving – is to think of the complex exponential as the 'complete' function, and sine and cosine as severed bits of it. No wonder they don't behave as nicely!