Trigonometric Polynomial Form: How Does the +1 Factor Fit in T_n(x)?

[quasar987]

Homework Statement

Maybe I'm just blind, but how is

$$T_n(x)=\left(\frac{1+\cos(t-a)}{b}\right)^n$$

of the form

$$T_n(x)=\sum_{k=-n}^nc_ke^{ikx}$$

?

I can get

$$\left(\frac{\cos(t-a)}{b}\right)^n$$

by setting

$$c_{\pm n}=\left(\frac{e^{\mp ia}}{2b}\right)^n$$

and the rest of the c_k= 0, but how does the +1 comes in?

 

[HallsofIvy]

][tex](1+ a)^n= 1+ a+ _nC_2a^2+ ...+ a^{n-1}+ a^n[/itex] by the binomial theorem. Now use what you give to write each term as an exponential.

 

[quasar987]

Ah!

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