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cis

cis Function

Syntax:

cis radians → number

Arguments and Values:

radians—a real.

number—a complex .

Description:

cis returns the value of ei· radians , which is a complex in which the real part is equal to the cosine of radians, and the imaginary part is equal to the sine of radians.

Examples:

(cis 0) → #C(1.0 0.0)

See Also:

Section 12.1.3.3 (Rule of Float Substitutability)

Expanded Reference: cis​

Basic usage​

cis returns e^(i*radians), a complex number whose real part is cos(radians) and whose imaginary part is sin(radians).

(cis 0)
=> #C(1.0 0.0)
(cis (/ pi 2))
;; => impl-dependent
(cis pi)
;; => impl-dependent

Euler's formula​

cis directly implements Euler's formula: e^(itheta) = cos(theta) + isin(theta).

;; Verify that cis matches cos + i*sin
(let ((theta 1.0))
  (list (cis theta)
        (complex (cos theta) (sin theta))))
=> (#C(0.5403023 0.84147096) #C(0.5403023 0.84147096))

Unit circle representation​

cis always returns a complex number on the unit circle (magnitude 1).

(abs (cis 0.0))
=> 1.0
(abs (cis 1.0))
=> 0.99999994
(abs (cis 42.0))
;; => impl-dependent

Constructing complex numbers in polar form​

Use cis together with multiplication to create complex numbers from polar coordinates (magnitude and angle).

;; Complex number with magnitude 5 and angle pi/4
(* 5 (cis (/ pi 4)))
;; => impl-dependent

;; Complex number with magnitude 2 and angle pi/3
(* 2 (cis (/ pi 3)))
;; => impl-dependent

Relationship to phase and abs​

For any complex number z, (* (abs z) (cis (phase z))) reconstructs z.

(let ((z #c(3.0 4.0)))
  (* (abs z) (cis (phase z))))
=> #C(3.0 4.0)