Help with derivation

[iocal]

Hi guys. I am looking for some help to understand the derivation of an important result.

|e^itx| = |cox(tx)+isin(tx)|=√|cos^2(tx)+sin^2(tx)|=1

i being the imaginary number. I understand Euler's formula, what I do not understand is the second to last result. Is it an application of De Moivres theorem or something else?
This by the way is the proof that the characteristic function in statistics always exists.

 

[pka]

Hi guys. I am looking for some help to understand the derivation of an important result.
|e^itx| = |cox(tx)+isin(tx)|=√|cos^2(tx)+sin^2(tx)|=1
i being the imaginary number. I understand Euler's formula, what I do not understand is the second to last result. Is it an application of De Moivres theorem or something else?
This by the way is the proof that the characteristic function in statistics always exists.

I assume that by "second to last result." you mean that.
It that is correct, then the answer can be as complicated as you like; it depends upon where we begin with basic concepts.

In general, we define \(\displaystyle \exp(z)=e^z=r\cos(\theta)+i \)\(\displaystyle r\sin(\theta) \) where \(\displaystyle \theta=\text{Arg}(z)\) and \(\displaystyle r=|z|\).

If I have misread your question, please correct us.

 

[stapel]

|e^itx| = |cox(tx)+isin(tx)|=√|cos^2(tx)+sin^2(tx)|=1

...what I do not understand is the second to last result. Is it an application of De Moivres theorem or something else?

Isn't this just an application of the definition of the absolute value of a complex number?

 

[HallsofIvy]

As stapel suggested this is just the absolute value of the complex number. If z= a+ bi, then \(\displaystyle |z|= \sqrt{a^2+ b^2}\). Here, a= cos(tx) and b= sin(tx) so that \(\displaystyle |cos(tx)+ i sin(tx)|= \sqrt{cos^2(tx)+ sin^2(tx)}= \sqrt{1}= 1\).

Another way of looking at is this: In the "complex plane" we associate the number x+ iy with the point (x, y). With, as here, x= cos(t), y= sin(t), we have \(\displaystyle x^2+ y^2= cos^2(t)+ sin^2(t)= 1\). That is, every such point on the unit circle. |z| is, geometrically, the distance from 0 to z- here, 1 for every point on the unit circle.

 

[iocal]

All of the above are very helpful. I am not proud of that but I had definitely missed the absolute value point which by the way is also the length of the vector in the Argand diagram.
Thanks guys!