Vocabulary/odot
>> << Down to: Dyad Back to: Vocabulary Thru to: Dictionary
-
o. y Pi Times
Rank 0 -- operates on individual atoms of y, producing a result of the same shape -- WHY IS THIS IMPORTANT?
Returns (π times y) given any number y .
o. 1 3.14159 o. 1r3 NB. π/3 (slightly above 1 rad) 1.0472 o. i.5 0 3.14159 6.28319 9.42478 12.5664
Common uses
1. Represent, in J, common physics expressions involving π
sin=: 1&o. NB. see below: dyadic (o.) pi=: o. : ([ * [: o. ]) r=: 10 pi 0 1 2 0 3.14159 6.28319 2 pi r 62.8319
You can also use J's 'p'-notation to accurately represent expressions involving π
1p1 NB. pi 3.14159 3p2 NB. 3 times pi-squared 29.6088 3* (1p1)^2 NB. (equiv) 29.6088
2. Convert radians <--> degrees
rfd=: 180 %~ o. NB. radians from degrees dfr=: rfd^:_1 NB. degrees from radians rfd 180 3.14159 dfr 1p1 180 dfr 0.5p1 90
Use These Combinations
Combinations using o. y that have exceptionally good performance include:
-
e^ π y^ ^@o. y handles large values of y
-
x o. y Circle Function
Rank 0 0 -- operates on individual atoms of x and y, producing a result of the same shape -- WHY IS THIS IMPORTANT?
Combines the common trigonometric and hyperbolic functions, and their inverses, without the need for reserved words like sin, cos, etc.
cop=: 0&o. NB. sqrt (1-(y^2)) sin=: 1&o. NB. sine of y cos=: 2&o. NB. cosine of y tan=: 3&o. NB. tangent of y coh=: 4&o. NB. sqrt (1+(y^2)) sinh=: 5&o. NB. hyperbolic sine of y cosh=: 6&o. NB. hyperbolic cosine of y tanh=: 7&o. NB. hyperbolic tangent of y conh=: 8&o. NB. sqrt -(1+(y^2)) real=: 9&o. NB. Real part of y magn=: 10&o. NB. Magnitude of y imag=: 11&o. NB. Imaginary part of y angle=: 12&o. NB. Angle of y arcsin=: _1&o. NB. inverse sine arccos=: _2&o. NB. inverse cosine arctan=: _3&o. NB. inverse tangent cohn=: _4&o. NB. sqrt (_1+(y^2)) arcsinh=: _5&o. NB. inverse hyperbolic sine arccosh=: _6&o. NB. inverse hyperbolic cosine arctanh=: _7&o. NB. inverse hyperbolic tangent nconh=: _8&o. NB. -sqrt -(1+(y^2)) same=: _9&o. NB. y conj=: _10&o. NB. complex conjugate of y jdot=: _11&o. NB. j. y expj=: _12&o. NB. ^ j. y
Common uses
1. To work with trigonometric functions.
2. 9 o. y (real) and 11 o. y (imag) are the best ways to extract the real and imaginary parts of y.
3. To manipulate screen graphics.
4. cop offers occasional convenience in modifying circle functions to work with complementary y
5. atan2 =: 12 o. j. gives the angle in the correct quadrant.
likewise coh for hyperbolic functions
cop leverages the identity: assert 1 -: (*: sin y) + (*: cos y) for all y
sin rfd 30 0.5 cop@sin rfd 60 0.5
5. Euler's Identity 0 = 1 + e^ i π^
1 + expj 1p1 0
Related Primitives
Real/Imag (+. y), Signum (Unit Circle) (* y), Length/Angle (*. y), Magnitude (| y), Imaginary * Complex (j.), Angle * Polar (r.)
More Information
1. The inverse of x o. y is (-x) o. y for _7 ≤ x ≤ 7.
2. As a mnemonic, the odd values of x correspond to odd functions and the even values of x to even functions.