22. Miscellaneous

Dirichlet kernel

The Dirichlet kernel D_n(x) is the function occurring on both sides of the next identity:

${\displaystyle 1+2\cos x+2\cos(2x)+2\cos(3x)+\cdots +2\cos(nx)={\frac {\sin \left[\left(n+{\frac {1}{2}}\right)x\right\rbrack }{\sin \left({\frac {x}{2}}\right)}}.}$

The convolution of any integrable function of period 2π with the Dirichlet kernel coincides with the function's nth-degree Fourier approximation. The same holds for any measure or generalized function.

Tangent half-angle substitution

If we set

${\displaystyle t=\tan {\frac {x}{2}},}$

then

${\displaystyle \sin x={\frac {2t}{1+t^{2}}};\qquad \cos x={\frac {1-t^{2}}{1+t^{2}}};\qquad e^{ix}={\frac {1+it}{1-it}}}$

where e^ix = cos x + i sin x, sometimes abbreviated to cis x.

When this substitution of t for tan x/2 is used in calculus, it follows that sin x is replaced by 2t/1 + t², cos x is replaced by 1 − t²/1 + t² and the differential dx is replaced by 2 dt/1 + t². Thereby one converts rational functions of sin x and cos x to rational functions of t in order to find their antiderivatives.

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