BUILT-IN MATHEMATICA SYMBOL

Im

Im[z]
gives the imaginary part of the complex number

.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

Im[expr] is left unevaluated if expr is not a numeric quantity.

Im automatically threads over lists.

EXAMPLES

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Basic Examples (3)

Find the imaginary part of a complex number:

Plot the imaginary part of a complex-valued function:

Use Im to specify regions of the complex plane:

Find the imaginary part of a complex number:

In[1]:=

Out[1]=

Plot the imaginary part of a complex-valued function:

In[1]:=

Out[1]=

Use Im to specify regions of the complex plane:

In[1]:=

Out[1]=

Scope (5)

Mixed-precision complex inputs:

Exact complex inputs:

Algebraic numbers:

Transcendental numbers:

Im threads element-wise over lists:

For some input Im will automatically simplify:

TraditionalForm formatting:

Generalizations & Extensions (1)

Infinite arguments give symbolic results:

Check that quantity is in the upper half-plane:

Applications (3)

Flow around a cylinder as the imaginary part of a complex-valued function:

Construct a bivariate harmonic function from a complex function:

The function satisfies Laplace's equation:

Reconstruct an analytic function

from its real part

:

Example reconstruction:

Check result:

Properties & Relations (7)

Use Simplify and FullSimplify to simplify expressions containing Im:

Prove that the disk

is in the upper half-plane:

ComplexExpand assumes variables to be real:

Here z is not assumed real, and the result should be in terms of Re and Im:

FunctionExpand does not assume variables to be real:

Use Im to describe regions in the complex plane:

Reduce can solve equations and inequalities involving Im:

With FindInstance you can get sample points of regions:

Use Im in Assumptions:

Integrate can generate conditions in terms of Im:

Possible Issues (1)

Im can stay unevaluated for numeric arguments:

Additional transformation may simplify it:

Neat Examples (1)

Use Im to plot a 3D projection of the Riemann surface of

: