PRELIMINARY:

1. Introduction (~ 10 pages)

1.1. The strong interactions before the Standard Model

1.2. Causality, unitarity and crossing symmetry

1.3. Standard dispersion relations

1.4. Pitfalls of analytic continuation

2. Modern approach to analyticity (~ 15 pages)

2.1. Oehme's argument for Quantum Chromodynamics (QCD)

2.2. Operator Product Expansion (OPE) and analytic continuation

2.3. Analyticity versus Chiral Perturbation Theory (ChPT) and lattice QCD

2.4. Phase and modulus constraints

2.5. Functional-analysis based approach

3. Optimization problems in complex analysis (~ 40 pages)

3.1. Extremal problems in L² norm

3.2. Extremal problems in L⁸ norm

3.3. Generalizations: vector-valued and non-real analytic functions

3.4. Series optimization by conformal mappings

4. Optimization methods for precision predictions in hadron physics (~ 50 pages)

4.1. Unitarity bounds for weak hadronic form factors

4.2. Electromagnetic charge radius of the pion at high precision

4.3. Virtual Compton scattering on proton

4.4. Conformal mappings in perturbative QCD

4.5. Testing quark-hadron duality violation

5. Other applications of analyticity (~ 10 pages)

5.1. Rigorous results from axiomatic field theory

5.2. Detection of broad resonances

5.3. Coupled-channels and three-body decays

5.4. Divergent series and hyperasymptotics

5.5. Outlook