6.23.3  Increasing the phase by π/2 in a trigonometric expression: shift_phase

The shift_phase command increases the phase of a trigonometric expression by π/2.

• shift_phase takes one argument:
  expr, a trigonometric expression.
• shift_phase(expr) returns expr with the phase increased by π/2 (after automatic simplification).

Examples.

• Input:
  shift_phase(x + sin(x))
  Output:

  x−cos

  ⎛ ⎜ ⎜ ⎝

  π +2 x 2

  ⎞ ⎟ ⎟ ⎠

• Input:
  shift_phase(x + cos(x))
  Output:

  x+sin

  ⎛ ⎜ ⎜ ⎝

  π +2 x 2

  ⎞ ⎟ ⎟ ⎠

• Input:
  shift_phase(x + tan(x))
  Output:

  x−	1 tan ⎛ ⎜ ⎜ ⎝ π +2 x 2 ⎞ ⎟ ⎟ ⎠

Quoting the argument will prevent the automatic simplification.

Example.
Input:

Output:

With an unquoted sine, you get:
Input:

Output:

since sin(x+pi/2) is evaluated (in this case simplified) before shift_phase is called, and shift_phase(cos(x)) returns sin((pi+2*x)/2).