"Bending up" $LC$-circuit to a linear antenna mathematically In introductory physics books one often finds a picture series leading from an $LC$-circuit to a simple linear antenna by "bending up" the $LC$-circuit, for example like this:

The key difference between the first and last picture is the described as that the fields and the $EM$-energy are very good localized in the $LC$-circuit but expanding in space in the right picture as indicated in the next figure:

In my own words, I would say that the $LC$-circuit will already emit an $EM$-wave, but with a power magnitudes smaller thatn the linear antenna. 


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*Are there any experimental measurements available which compair (in far field) the power radiated by the $LC$-circuit with the linear antenna such that I can get a feeling of how much magnitudes the difference will be.

*Can this deformation be made mathematically more rigorous?
 A: 
Are there any experimental measurements available which compair (in far field) the power radiated by the LC circuit

The radiation coming from a discrete inductor or capacitor will depend on the details of their construction. For example, it's possible to buy a "shielded" inductor which has a ferrite material surrounding it as well as in its core, for the purpose of reducing radiation from the inductor. 
The radiation coming from a tank circuit made of a discrete inductor and capacitor is likely to come more from the wires connecting the parts rather than from the parts themselves, and it will depend on the details of the construction of the circuit. How long are the wires, and how far apart are they, for example.
So there's no way to compare a "generic" discrete-element LC circuit with an equivalent antenna the way you're asking about. 

Can this deformation made mathematically more rigorous?

You can use finite element analysis to analyze an antenna and find the equivalent L and C elements to model it as a lumped circuit.
However I don't think this is more "rigorous" in the way most physicists use the term "rigorous".
