  Additivity of Breit-Wigner phases has been proposed to describe interfering
resonances in partial waves in $\pi\pi$ scattering. This assumption leads to an
expression for partial wave amplitudes that involves products of Breit-Wigner
amplitudes. We show that this expression is equivalent to a coherent sum of
Breit-Wigner amplitudes with specific complex coefficients which depend on the
resonance parameters of all contributing resonances. We use analyticity of
$\pi\pi$ partial wave amplitudes to show that they must have the form of a
coherent sum of Breit-Wigner amplitudes with complex coefficients and a complex
coherent background. The assumption of additivity of Breit-Wigner phases
restricts the partial waves to analytical functions with very specific form of
residues of Breit-Wigner poles. We argue that the general form provided by the
analyticity is more appropriate in fits to data to determine resonance
parameters. The partial wave unitarity can be imposed using the modern methods
of constrained optimization. We discuss unitarity and the production amplitudes
in $\pi N\to\pi\pi N$ and use analyticity in the dipion mass variable to
justify the common practice of writing the production amplitudes as a coherent
sum of Breit-Wigner amplitudes with free complex coefficients and a complex
coherent background in fits to mass spectra with interfering resonances.
