Editing Hartley transform (section)

=== cas ===
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The properties of the ''Hartley kernel'', for which Hartley introduced the name ''cas'' for the function (from ''cosine and sine'') in 1942,<ref name="Hartley_1942"/><ref name="Bracewell_1999"/> follow directly from [[trigonometry]], and its definition as a phase-shifted trigonometric function {{nowrap|1=<math>\operatorname{cas}(t)=\sqrt{2} \sin (t+\pi /4)=\sin(t)+\cos(t)</math>.}} For example, it has an angle-addition identity of:

<math display=block>
2 \operatorname{cas} (a+b) = \operatorname{cas}(a) \operatorname{cas}(b) + \operatorname{cas}(-a) \operatorname{cas}(b) + \operatorname{cas}(a) \operatorname{cas}(-b) - \operatorname{cas}(-a) \operatorname{cas}(-b)\,.
</math>

Additionally:

<math display=block> 
\operatorname{cas} (a+b) = {\cos (a) \operatorname{cas} (b)} + {\sin (a) \operatorname{cas} (-b)} = \cos (b) \operatorname{cas} (a) + \sin (b) \operatorname{cas}(-a)\,,
</math>

and its derivative is given by:

<math display=block>
\operatorname{cas}'(a) = \frac{d}{da} \operatorname{cas} (a) = \cos (a) - \sin (a) = \operatorname{cas}(-a)\,.
</math>