Im¶

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description¶

Im[z] gives the imaginary part of numeric z, and 0 for real or real-valued (Re/Im/Abs/Arg) arguments.

Examples¶

No verified examples yet for this function.

Implementation notes¶

builtin_im returns the imaginary part. It returns 0 for the real-valued-by-construction heads (Re/Im/Abs/Arg) and for any real numeric kind (Integer/Real/Rational/MPFR), copies the second component of a Complex[re, im] literal, and for a general expression runs complex_decompose (a recursive Plus/Times walk that propagates Complex literals through complex multiplication) — returning the imaginary part only when both decomposed parts are concretely numeric (is_numeric_real). Otherwise NULL, leaving the symbolic head in place. Re/ReIm in the same file share this machinery.

Attributes: Listable, NumericFunction, Protected.

Implementation status¶

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References¶

Source: src/complex.c
Specification: docs/spec/builtins/arithmetic.md

Notes & additional examples¶

Worked examples¶

In[1]:= Im[3 + 4 I]
Out[1]= 4

In[2]:= Im[7]
Out[2]= 0

Im resolves exact algebraic and transcendental values, including radicals of negative numbers and branch-cut logarithms:

In[1]:= Im[Sqrt[-4]]
Out[1]= 2

In[2]:= Im[(1 + I)^10]
Out[2]= 32

In[3]:= Im[Log[-1]]
Out[3]= Pi

Being Listable, it threads over a list of numbers:

In[1]:= Im[{1 + 2 I, 3 - 4 I, 5}]
Out[1]= {2, -4, 0}

For values it cannot reduce in closed form, Im stays symbolic but still yields to high-precision numerics — here the imaginary part of Γ(1 + i):

In[1]:= Im[Gamma[1 + I]]
Out[1]= Im[Gamma[1 + I]]

In[2]:= N[Im[Gamma[1 + I]], 40]
Out[2]= -0.15494982830181068512495513048388660519589

Notes¶

Im[z] extracts the imaginary part of numeric z, giving 0 for real (or real-valued) arguments. It is Listable.