page_title: Exponential integral
section_title: Exponential integral of imaginary argument
hierarchical_section_title: Exponential integral / Properties / Exponential integral of imaginary argument
caption_attribution_description: w:Exponential integral of imaginary argument, E 1 ( i &#xA0; x ) {\displaystyle {\rm {E}}_{1}({\rm {i}}\!~x)} versus &#xA0; x &#xA0; {\displaystyle ~x~} , real part (black) and imaginary part(red)
context_page_description: In mathematics, the exponential integral Ei is a special function on the complex plane.
  It is defined as one particular definite integral of the ratio between an exponential function and its argument.
context_section_description: If  is imaginary, it has a nonnegative real part, so we can use the formula
  to get a relation with the trigonometric integrals  and :
  The real and imaginary parts of  are plotted in the figure to the right with black and red curves.
caption: E
            
              1
            
          
          (
          i
          x
          )
        
      
      {\displaystyle E_{1}(ix)}
    
   against 
    
      
        
          x
        
      
      {\displaystyle x}
    
  ; real part black, imaginary part red.
exif: {}