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*The [[Fourier transform]] of a purely real-valued even function is real and even. (see {{slink|Fourier_analysis|Symmetry_properties}})
*The Fourier transform of a purely real-valued odd function is imaginary and odd. (see {{slink|Fourier_analysis|Symmetry_properties}})
==In computing==
In the [[APL (programming language)|APL programming language]], all [[trigonometric functions]], and often several [[complex analysis]] functions, are combined into a single [[binary function]] (called a ''dyadic function'') which, per [[APL_syntax_and_symbols#Monadic_and_dyadic_functions|APL syntax]], uses [[infix notation]]. This function, denoted by the Unicode character {{unichar|25cb|WHITE CIRCLE}}, takes one [[parameter (computer programming)|parameter]] as a function selector (denoted <source lang=apl inline>X</source> in the below table), while the other (<source lang=apl inline>Y</source>) is the selected function's parameter. The function selector is an integer usually in the [[interval (mathematics)|interval]] ⟦−12, 12⟧<ref>Dyalog Ltd. Primitive Functions: [https://help.dyalog.com/latest/#Language/Primitive%20Functions/Circular.htm Trigonometric Functions]. Retrieved 2022-07-07.</ref> or ⟦−7, 7⟧<ref>MicroAPL Ltd. Primitives: [https://microapl.com/apl_help/ch_020_020_240.htm APL <source lang=apl inline>○</source> Circular and Hyperbolic functions]. [[APLX]] Help: Help on APL language. Retrieved 2022-07-07.</ref> depending on whether the implementation in question supports [[complex number]]s or not. Each function selector has been chosen such that its [[Parity (mathematics)|parity]] matches that of its function, and its [[Negative number|negation]] selects its [[inverse function]], with the function selected by 0 being its own inverse.<ref>[[Eugene McDonnell|McDonnell, Eugene]]. [https://www.jsoftware.com/papers/eem/storyofo.htm The Story of <source lang=apl inline>○</source>]. APL Quote-Quad. Volume 8. Number 2. 1977-12.</ref> The most common definitions are:<ref>APL Wiki. [https://aplwiki.com/wiki/Circular Circular]. Retrieved 2022-07-07.</ref>
{| class="wikitable c" style="margin: 1em auto 1em auto;text-align: center;"
! style="width:33%" | <source lang=apl inline>(-X)○Y</source> !! style="width:33%" | <source lang=apl inline>X</source> !! <source lang=apl inline>X○Y</source>
|-
| <math>\sqrt{1-Y^2}</math> || 0 || <math>\sqrt{1-Y^2}</math>
|-
| <math>\sin^{-1}{Y}</math> || 1 || <math>\sin{Y}</math>
|-
| <math>\cos^{-1}{Y}</math> || 2 || <math>\cos{Y}</math>
|-
| <math>\tan^{-1}{Y}</math> || 3 || <math>\tan{Y}</math>
|-
| <math>\sqrt{Y^2-1}</math> || 4 || <math>\sqrt{1+Y^2}</math>
|-
| <math>\sinh^{-1}{Y}</math> || 5 || <math>\sinh{Y}</math>
|-
| <math>\cosh^{-1}{Y}</math> || 6 || <math>\cosh{Y}</math>
|-
| <math>\tanh^{-1}{Y}</math> || 7 || <math>\tanh{Y}</math>
|}
==Harmonics==
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