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The slope of the exponential as a function of energy is then: Γ(E) = 1.45+E/4.87GeV. At the middle of the highest energy band this is Γ(31.6GeV)=7.94. A power law with this slope and integral flux given above has a flux constant at the middle of the energy band of N0N_0_=1.25e-09*(7.94-1)/31.6/((10/31.6)(1-7.94)-(100/31.6)(1-7.94))=9.35e-14 ph/cm^2/s/GeV, i.e.
- dN/dE = 9.35e-14 (E/31.6GeV)^-7.94 ph/cm^2/s/GeV
This power law is depicted on my plot as a red power law covering the range 10-100 GeV (the PL fits from the other bands are also shown). Numerically integrating the red PL function also gives 1.25e-09. Calculating E^2dN/dE at the mid point gives 31.6^2*9.3e-14 = 9.3e-11 GeV/cm^2/s = 1.5 erg/cm^2/s, in agreement with the point depicted on the plot.
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