## Online material

### Appendix A: Fitting functions

The Comptonized model (COMP) is a power-law model with a high-energy exponential cutoff: (A.1)where *A* is the normalization factor at 100 keV in units of ph s^{-1} cm^{-2} keV^{-1}, *α* is the power-law index, and *E*_{p} is the peak energy in the *ν**F*_{ν} space in units of keV.

The Band function (BAND) is a model which a low-energy cutoff power law and a high-energy power law joined together by a smooth transition. It is an empirical function proposed by Band et al. (1993): (A.2)where (A.3)In Eqs. (A.2) and (A.3), *A* is the normalization factor at 100 keV in units of ph s^{-1} cm^{-2} keV^{-1}, *α* is the low-energy power-law index, *β* is the high-energy power-law index, *E*_{p} is the peak energy in the *ν**F*_{ν} space in units of keV, and *E*_{c} is the characteristic energy where the low-energy power law with an exponential cutoff

ends and the pure high-energy power law starts, in units of keV. We note that when *β* → −∞ the Band function reduces to the Comptonized model.

The smoothly broken power law (SBPL) is a model of two power laws joined by a smooth transition. It was first parameterized by Ryde (1999) and then re-parameterized by Kaneko et al. (2006): (A.4)where (A.5)In Eqs. (A.4) and (A.5), *A* is the normalization factor at 100 keV in units of ph s^{-1} cm^{-2} keV^{-1}, *α* and *β* are the low- and high-energy power-law indices respectively, *E*_{b} is the break energy in units of keV, and Δ is the break scale. Unlike the Band function, the break scale is not coupled to the power-law indices, so SBPL is a five-parameter-model if we let Δ free to vary. It is fixed at Δ = 0.3 in all the *Fermi *GBM GRB catalogs and is therefore adopted in this paper.

The peak energy of SBPL in the *ν**F*_{ν} space can be found at (A.6)We note that Eq. (A.6) is only valid for *α*> −2 and *β*< −2.

### Appendix B: Comparison between the average time-resolved and time-integrated sharpness angles

Comparison between the average time-resolved and the time-integrated sharpness angles.

*© ESO, 2015*