Complex Differentiation sin(z) & cos(z)

[ryanj123]

Hi There,

I'm currently using the definition of exponential functions:

e^z=(e^x)(e^iy)

I need help defining:

sin(z)=(e^(iz)-e(-iz))/(2i)
cos(z)=(e^(iz)+e(-iz))/(2)

And showing that

sin'(z) = cos(z)
cos'(z) = -sin(z)
(sin(z))^2+(cos(z))^2=1

Any help would be appreciated. I'm thinking the CR equations need to be used, but it's just the algebraic manipulation I'm having problems with.

Thanks.

 

[Office_Shredder]

Do you know what the complex derivative of the exponential function is? Then you can just use the linearity of the derivative to calculate the derivative of sin(z) directly from its exponential form

 

[ryanj123]

I actually do not know the derivative of the exponential function.

 

[Hurkyl]

Well, let's start with that then. One of the main reasons to define sin(z) and cos(z) in that fashion is to take advantage of what we know about exp(z) -- so we should learn about exp(z) first!


The beginning is often a good place to start. What is the definition of exp(z)?

I suppose you said that you're taking exp(z) = e^x e^iy. (where, I assume, you're equating z with x + iy)

Well, what is e^x? What is e^iy? And how does one compute the complex derivative of a function when you've broken the argument into real and imaginary pieces?

 

[ryanj123]

Well, let's start with that then. One of the main reasons to define sin(z) and cos(z) in that fashion is to take advantage of what we know about exp(z) -- so we should learn about exp(z) first!


The beginning is often a good place to start. What is the definition of exp(z)?

I suppose you said that you're taking exp(z) = e^x e^iy. (where, I assume, you're equating z with x + iy)

Well, what is e^x? What is e^iy? And how does one compute the complex derivative of a function when you've broken the argument into real and imaginary pieces?

Well thinking about it, the exponential function is
e^z=(e^x)(e^iy)=(e^x)( cos(y)+i(sin(y)) ).
It's derivative I'm guessing is the four partial's where
u(x,y) = e^x(cos(y)) and,
v(x,y) = e^x(sin(y)).
Then ux=vy
and uy=-vx

So with this, must we turn e^iz into cos(z)+isin(z)?
Then differentiating with respect to Z, I'm again confused, do we turn this into z = x + iy?

 

[Hurkyl]

It's derivative I'm guessing

Don't guess! Surely your textbook has something to say on the issue?

(Note that the guess you actually made doesn't really make sense...)

Guessing has its place in mathematics. Guessing at the definition of an established technical term is not one of those places, particularly when it's definition is readily available.