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conjugate

conjugate Function

Syntax:

conjugate number → conjugate

Arguments and Values:

number—a number .

conjugate—a number .

Description:

Returns the complex conjugate of number. The conjugate of a real number is itself.

Examples:

(conjugate #c(0 -1)) → #C(0 1) 
(conjugate #c(1 1)) → #C(1 -1) 
(conjugate 1.5) → 1.5 
(conjugate #C(3/5 4/5)) → #C(3/5 -4/5) 
(conjugate #C(0.0D0 -1.0D0)) → #C(0.0D0 1.0D0) 
(conjugate 3.7) → 3.7

Notes:

For a complex number z,

(conjugate z) ≡ (complex (realpart z) (- (imagpart z)))

Expanded Reference: conjugate​

Basic complex conjugate​

conjugate negates the imaginary part of a complex number, leaving the real part unchanged.

(conjugate #c(3 4))
=> #C(3 -4)
(conjugate #c(0 -1))
=> #C(0 1)
(conjugate #c(1 1))
=> #C(1 -1)
(conjugate #c(3/5 4/5))
=> #C(3/5 -4/5)

Conjugate of real numbers​

The conjugate of a real number is itself.

(conjugate 5)
=> 5
(conjugate 1.5)
=> 1.5
(conjugate 3.7)
=> 3.7
(conjugate -2/3)
=> -2/3

Double-float complex conjugate​

(conjugate #c(0.0d0 -1.0d0))
=> #C(0.0d0 1.0d0)
(conjugate #c(2.5d0 3.5d0))
=> #C(2.5d0 -3.5d0)

Computing magnitude using conjugate​

The product of a complex number and its conjugate equals the square of the magnitude.

(let ((z #c(3.0 4.0)))
  (* z (conjugate z)))
=> #C(25.0 0.0)

(let ((z #c(3.0 4.0)))
  (sqrt (realpart (* z (conjugate z)))))
=> 5.0

Equivalence definition​

conjugate is equivalent to constructing a new complex from the same real part and the negated imaginary part.

(let ((z #c(7 -3)))
  (eql (conjugate z)
       (complex (realpart z) (- (imagpart z)))))
=> T