What is the list of circle functions in APL?

Question

According to http://www.microapl.co.uk/apl/APL1_2.PDF, there are circle function between ¯12 and 12. For example, functions 1, 2, and 3 are respectively sin, cos, and tan. I found on this reference what are functions ¯7 to 7. However, I didn't find what are functions 8 to 12 (and their reciprocals). Could anybody point me what they are?

Answer 1

I've looked at the reference of APL X, but I can not find a description of the left arguments, which allows a maximum range of -12 to 12. Where did you read that there is this range of values​​?

Dyalog APL allows a range from -12 to 12, here's a quote from the Dyalog APL Reference:

R ← X ○ Y  

Y must be numeric. X must be an integer in the range ¯ 12 ≤ X ≤ 12 R is numeric.

Perhaps you are confusing APL X and Dyalog APL.

These are the (Dyalog) operations for each value of X

(-X) ○ Y            |  X |  X ○ Y  
--------------------+----+--------------
(1-Y*2)*.5          |  0 | (1-Y*2)*.5 
Arcsin Y            |  1 | Sine Y 
Arccos Y            |  2 | Cosine Y 
Arctan Y            |  3 | Tangent Y 
(Y+1)×((Y-1)÷Y+1)*.5|  4 | (1+Y*2)*.5 
Arcsinh Y           |  5 | Sinh Y 
Arccosh Y           |  6 | Cosh Y 
Arctanh Y           |  7 | Tanh Y 
-8○Y                |  8 | (-1+Y*2)*0.5 
Y                   |  9 | a 
+Y                  | 10 | |Y 
b×0J1               | 11 | b 
*Y×0J1              | 12 | θ 

X determines which of a family of trigonometric, hyperbolic, Pythagorean and complex functions to apply to Y, from the following table.

Note that when Y is complex, a and b are used to represent its real and imaginary parts, while θ represents its phase.

Answer 2

In his "Dictionary of APL" Iverson defined even more, from -15 to 15: http://www.jsoftware.com/papers/APLDictionary1.htm#circle