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# Im

 Im[z]gives the imaginary part of the complex number .
• 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.
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:
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Plot the imaginary part of a complex-valued function:
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Use Im to specify regions of the complex plane:
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 Scope   (5)
Mixed-precision complex inputs:
Exact complex inputs:
Algebraic numbers:
Transcendental numbers:
For some input Im will automatically simplify:
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:
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:
Im can stay unevaluated for numeric arguments: