Faddeeva function

The Faddeeva function or Kramp function is a scaled <a href="/facts/Complex_number/2w7mqI7e">complex</a> complementary <a href="/facts/Error_function/Ca2H7pGr">error function</a>,

w
        (
        z
        )
        :=
        
          e
          
            −
            
              z
              
                2
              
            
          
        
        erfc
        ⁡
        (
        −
        i
        z
        )
        =
        erfcx
        ⁡
        (
        −
        i
        z
        )
        =
        
          e
          
            −
            
              z
              
                2
              
            
          
        
        
          (
          
            1
            +
            
              
                
                  2
                  i
                
                
                  π
                
              
            
            
              ∫
              
                0
              
              
                z
              
            
            
              e
              
                
                  t
                  
                    2
                  
                
              
            
            
              d
            
            t
          
          )
        
        .
      
    
    {\displaystyle w(z):=e^{-z^{2}}\operatorname {erfc} (-iz)=\operatorname {erfcx} (-iz)=e^{-z^{2}}\left(1+{\frac {2i}{\sqrt {\pi }}}\int _{0}^{z}e^{t^{2}}{\text{d}}t\right).}

It is related to the <a href="/facts/Fresnel_integral/5W7z760j">Fresnel integral</a>, to <a href="/facts/Dawson%2527s_integral/V89wVW9g">Dawson's integral</a>, and to the <a href="/facts/Voigt_function/uwQXjc4k">Voigt function</a>.
The function arises in various physical problems, typically relating to electromagnetic responses in complicated media.

<ul><li>problems involving small-amplitude waves propagating through <a href="/facts/Maxwell%25E2%2580%2593Boltzmann_distribution/KAuVfdKJ">Maxwellian</a> <a href="/facts/Plasma_(physics)/1d3btMVU">plasmas</a>, and in particular appears in the plasma's <a href="/facts/Permittivity/Sd3WtgT1">permittivity</a> from which <a href="/facts/Dispersion_relation/oo7RM5eg">dispersion relations</a> are derived, hence it is sometimes referred to as the plasma dispersion function (although this name is sometimes used instead for the rescaled function Z(z) = i√π w(z) defined by Fried and Conte, 1961).</li>
<li>the infrared <a href="/facts/Permittivity/Sd3WtgT1">permittivity</a> functions of amorphous oxides have resonances (due to <a href="/facts/Phonon/tflWW6vw">phonons</a>) that are sometimes too complicated to fit using simple harmonic oscillators. The <a href="/facts/Brendel%25E2%2580%2593Bormann_oscillator_model/3G9rVwMq">Brendel–Bormann oscillator model</a> uses an infinite superposition of oscillators having slightly different frequencies, with a Gaussian distribution. The integrated response can be written in terms of the Faddeeva function.</li>
<li>the Faddeeva function is also used in the analysis of electromagnetic waves of the type used in AM radio. Groundwaves are vertically polarised waves propagating over a lossy ground with finite resistivity and permittivity.</li>
<li>the Faddeeva function also describes the changes of the <a href="/facts/Neutron_cross_section/gAkJmXXA">neutron cross sections</a> of materials as temperature is varied.</li></ul>

Faddeeva function open-in-new

Faddeeva function