Find the imaginary part of a complex number:

Find the imaginary part of a complex number expressed in polar form:

Plot over a subset of the complex plane:

Use Im to specify regions of the complex plane:

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:

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:

ReImPlot plots the real and imaginary parts of a function:

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:

Im can stay unevaluated for numeric arguments:

Additional transformation may simplify it:

Im is a function of a complex variable and is therefore not differentiable:

As a complex function, it is not possible to write Im[z] without involving Conjugate[z]:

In particular, the limit that defines the derivative is direction dependent and therefore does not exist:

Use ComplexExpand to get differentiable expressions for real-valued variables:

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