complex

LFE support for numbers both real and imagined

Table of Contents

• Introduction
• Installation
• Usage
  • Creating Complex Numbers
    • From Strings
    • From Atoms
  • Convenience Functions
    • Printing
    • Common Numbers
  • Math
    • Arithmatic
    • Operations
    • Powers
• API
• License

Introduction ↟

This library provides complex number data types (LFE records for rectangular and polar complex numbers) as well as many mathematical operations which support the complex data type. For a full list of functions in the API,see the bottom of this README file.

Installation ↟

Just add it to your rebar.config deps:

{deps, [
    {complex, {git, "https://github.com/lfex/complex.git", {branch, "master"}}}
  ]}.

At this point, the complex library will be avialable in your project's dependencies.

Usage ↟

Creating Complex Numbers ↟

Create some new complex numbers using the standard rectangular coordinates:

lfe> (set z1 (complex:new 4 -2))
#(complex 4 -2)
lfe> (set z2 (complex:new 4 2))
#(complex 4 2)

Create complex numbers using polar coordinates:

lfe> (set z3 (complex:new-polar 4 (* -0.5 (math:pi))))
#(complex-polar 4 -1.5707963267948966)
lfe> (set z4 (complex:new-polar 4 (* 0.5 (math:pi))))
#(complex-polar 4 1.5707963267948966)

Creating from Strings ↟

You can also create a new complex number using a string value:

lfe> (complex:new "4-2i")
#(complex 4 -2)
lfe> (complex:new "4+2i")
#(complex 4 2)

There are rules for using complex strings, though:

• there can be no spaces in the complex string
• you must always include the real part, even if the value is zero: (complex:new "0+2i")
• the imaginary part must always include the number, even if the component value is 1: (complex:new "2+1i")

Optional usage:

• if the imaginary component is zero, you may leave it off: (complex:new "2")
• you may use i, j, I, or J to indicate the imaginary part: (complex:new "-4+2j")
• you may use floating point values: (complex:new "1.2-3.4i")
• you may use scientific notation: (complex:new "1.2e3-4.5e-6i")

Creating from Atoms ↟

Using the same rules, you may use atoms to create a new complex number:

lfe> (complex:new '4-2i)
#(complex 4 -2)
lfe> (complex:new '4.3+2.1i)
#(complex 4.3 2.1)
lfe> (complex:new '4.3e10-2.1e-20j)
#(complex 4.3e10 -2.1e-20)

However, do keep in mind that the use of atoms to create complex numbers should not be done automatically in large numbers, or you run the risk of exhuasting the Erlang atom table and thus crashing your VM.

Convenience Functions ↟

For the rest of the usage, we'll just slurp so that the calls are easier to type:

lfe> (slurp "src/complex.lfe")
#(ok complex)

Printing Complex Numbers ↟

Print the numbers we previously defined:

lfe> (format z1)
4-2i
ok
lfe> (format z2)
4+2i
ok
lfe> (format z3)
0-4.0i
ok
lfe> (format z4)
0+4.0i
ok

Note that format/1 will convert a polar coordinate to rectangular; thus the last two above.

Common Numbers ↟

lfe> (one)
#(complex 1 0)
lfe> (two)
#(complex 2 0)
lfe> (i)
#(complex 0 1)
lfe> (pi)
#(complex 3.141592653589793 0)
lfe> (e)
#(complex 2.718281828459045 0)
lfe> (-pi/2)
#(complex -1.5707963267948966 0)

Math ↟

Arithmatic ↟

lfe> (add (complex 4 2) (i))
#(complex 4 3)
lfe> (sub (complex 4 2) (i))
#(complex 4 1)
lfe> (mult (complex 4 2) (i))
#(complex -2 4)
lfe> (div (complex 4 2) (i))
#(complex 2.0 -4.0)

Note that complex/2 is an alias for new/2; it just looks nicer when not using the module name.

Operations ↟

lfe> (conj z2)
#(complex 4 -2)
lfe> (eq z1 z2)
false
lfe> (eq z1 (conj z2))
true
lfe> (inv z1)
#(complex 0.2 0.1)
lfe> (inv z2)
#(complex 0.2 -0.1)
lfe> (modulus z1)
4.47213595499958
lfe> (modulus z1 #(complex))
#(complex 4.47213595499958 0)
lfe> (modulus (complex-polar 4 (math:pi)))
4
lfe> (arg (complex-polar 4 (math:pi)))
3.141592653589793

lfe> (sqrt (-one))
#(complex 0.0 1.0)
ok
lfe> (eq (sqrt (-one)) (i))
true

Powers ↟

Using exponents to demonstrate the cyclic values of the powers of i:

lfe> (format (pow (i) 0))
1+0i
ok
lfe> (format (pow (i) 1))
0+1i
ok
lfe> (format (pow (i) 2))
-1+0i
ok
lfe> (format (pow (i) 3))
0-1i
ok
lfe> (format (pow (i) 4))
1+0i
ok

Negative powers are supported:

lfe> (pow (pi) -2)
#(complex 0.10132118364233778 0.0)
lfe> (pow (two) -4)
#(complex 0.0625 0.0)

As are fractional powers (roots):

lfe> (pow 16 (/ 1 2))
#(complex 4.0 0)
lfe> (pow 16 (/ 1 4))
#(complex 2.0 0)
lfe> (pow 16 (/ 1 8))
#(complex 1.4142135623730951 0)

See the unit tests for a greater number of examples.

API ↟

The list of functions currently supported by the complex library are as follows:

complex:-2pi/0
complex:->atom/1
complex:->str/1
complex:-i/0
complex:-i/2/0
complex:-one/0
complex:-pi/0
complex:-pi/2/0
complex:-two/0
complex:2pi/0
complex:abs/1
complex:abs/2
complex:acos/1
complex:acosh/1
complex:acot/1
complex:acoth/1
complex:acsc/1
complex:acsch/1
complex:add/2
complex:angle/1
complex:arg/1
complex:arg/2
complex:asec/1
complex:asech/1
complex:asin/1
complex:asinh/1
complex:atan/1
complex:atanh/1
complex:atom->/1
complex:complex/2
complex:complex-polar/2
complex:complex-polar?/1
complex:complex?/1
complex:conj/1
complex:cos/1
complex:cosh/1
complex:cot/1
complex:coth/1
complex:csc/1
complex:csch/1
complex:distance/1
complex:div/2
complex:e/0
complex:eeq/2
complex:eq/2
complex:eq/3
complex:exp/1
complex:i/0
complex:i/2/0
complex:img/1
complex:inv/1
complex:ln/1
complex:modsq/1
complex:modulus/1
complex:modulus/2
complex:mult/2
complex:neg/1
complex:new/0
complex:new/1
complex:new/2
complex:new-polar/0
complex:new-polar/1
complex:new-polar/2
complex:one/0
complex:phase/1
complex:phi/1
complex:pi/0
complex:pi/2/0
complex:polar->rect/1
complex:polar->rect/2
complex:pow/2
complex:print/1
complex:r/1
complex:real/1
complex:rect->polar/1
complex:sec/1
complex:sech/1
complex:sign/1
complex:sin/1
complex:sinh/1
complex:sqrt/1
complex:str->/1
complex:sub/2
complex:tan/1
complex:tanh/1
complex:two/0

License ↟

Apache Version 2 License

Copyright © 2015-2020, Duncan McGreggor oubiwann@gmail.com