Usually, people use Schechter parameterization to show the luminosity functions.
From WIKI:
Schechter luminosity function[edit]
The Schechter luminosity function provides a parametric description of the space density of galaxies as a function of their luminosity. The form of the function is
where 
, and 
is a characteristic galaxy luminosity where the power-law form of the function cuts off. The parameter 
has units of number density and provides the normalization. The galaxy luminosity function may have different parameters for different populations and environments; it is not a universal function. One measurement from field galaxies is 
.[2]
, and is a characteristic galaxy luminosity where the power-law form of the function cuts off. The parameter has units of number density and provides the normalization. The galaxy luminosity function may have different parameters for different populations and environments; it is not a universal function. One measurement from field galaxies is .[2]
It is often more convenient to rewrite the Schechter function in terms of magnitudes, rather than luminosities. In this case, the Schechter function becomes:
![n(M) \ \mathrm{d}M = 0.4 \ \ln 10 \ \phi^* [ 10^{ -0.4 ( M - M^* ) } ]^{ \alpha + 1} \exp [ -10^{ -0.4 ( M - M^* ) } ] \ \mathrm{d}M .](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_slArR2M1VKlqFnnQ7zEJh6xFc1NPaqwK2itsye7yYZP_XThV4fS9fqS-mQnKfakRMr7U9IGu8_WFjDUn_hnDw5CKs9_M21MusXOQOdQZMsH5KCoCGi-uKUVHtdtveOnh_kr5SAZcXXtaq4iu7miA=s0-d)
Note that because the magnitude system is logarithmic, the power law has logarithmic slope 
. This is why a Schechter function with 
is said to be flat.
. This is why a Schechter function with is said to be flat.
Notice, in the papers, people use a instead of a+1.
Some reference:
(1) In python. you can calculate the n(M) using the parameters of z=1.5 from Oesch et al. 2010
(2) More star formation may occur in even lower-mass
galaxies having higher EWHα and that dominate the
UV luminosity function at z ∼ 2 (Fumagalli et al. 2012;
Alavi et al. 2013).
Fumagalli et al. 2012 Hα Equivalent Widths from the 3D-HST Survey: Evolution with Redshift and Dependence on Stellar Mass
Alavi et al. 2013 Ultra-faint Ultraviolet Galaxies at z ~ 2 behind the Lensing Cluster A1689: The Luminosity Function, Dust Extinction, and Star Formation Rate Density
1 2 3 4 5 | phi = -2.64 # normalization M = -19.82 # the standard luminosity a = -1.46 # slope X= M # the luminosity you want to calculate 10**phi*(np.log(10)/2.5) * ((10**(-0.4*(X-M)))**(a)) * np.e**(-10**(-0.4*(X-M))) |
