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○ — Cis / Circular

Function. Keys: Alt-o. Hanzi: 圆 (圓). Ranks: 0 monadic, 0 0 dyadic

○Y: eiY = cos Y + i sin Y. Real Y is an angle in radians. Pervasive.

○0                ⍝ 1
○0j1              ⍝ *¯1

K○Y applies the circular function with code K, from 0 to 8. K○⁻Y applies its inverse. Pervasive; angles in radians.

1○0               ⍝ 0
2○0               ⍝ 1
1○⁻1             ⍝ 1.5707963267948966
k k○y k○⁻y
0 √(1−y²) √(1−y²)
1 sin arcsin
2 cos arccos
3 tan arctan
4 √(1+y²) APL branch of √(y²−1)
5 sinh arcsinh
6 cosh arccosh
7 tanh arctanh
8 √(-1−y²) -√(-1−y²)

Uses complex continuation where needed. The cuts of 3○⁻ and 5○⁻ lie on the imaginary axis. The signed real zeros in 3○⁻ 0j2 and 3○⁻ ¯0j2 distinguish the two sides. Zero’s sign is not guaranteed. Under dynamic kinds, the literal 2j¯0 is read as real 2. A real argument computes with an imaginary part of +0:

1○⁻2              ⍝ 1.5707963267948966j1.3169578969248166

APL difference: there are no negative codes or codes 9 to 12. Write ¯k○Y as k○⁻Y. ∨Y gives the real and imaginary parts, |Y the magnitude, and ∧Y the magnitude and phase. On a branch cut, APL takes the side below the cut for a positive real argument.

Inverse

○⁻Y gives the angle of the unit complex number Y. The table above gives the inverse of each K○.

○⁻0j1   ⍝ π0.5