Editing Exponential integral (section)

===Relation with other functions===
Kummer's equation
:<math>z\frac{d^2w}{dz^2} + (b-z)\frac{dw}{dz} - aw = 0</math>
is usually solved by the [[confluent hypergeometric functions]] <math>M(a,b,z)</math> and <math>U(a,b,z).</math> But when <math>a=0</math> and <math>b=1,</math> that is,
:<math>z\frac{d^2w}{dz^2} + (1-z)\frac{dw}{dz} = 0</math>
we have
:<math>M(0,1,z)=U(0,1,z)=1</math>
for all ''z''. A second solution is then given by E<sub>1</sub>(−''z''). In fact,
:<math>E_1(-z)=-\gamma-i\pi+\frac{\partial[U(a,1,z)-M(a,1,z)]}{\partial a},\qquad 0<{\rm Arg}(z)<2\pi</math>
with the derivative evaluated at <math>a=0.</math> Another connexion with the confluent hypergeometric functions is that ''E<sub>1</sub>'' is an exponential times the function ''U''(1,1,''z''):
:<math>E_1(z)=e^{-z}U(1,1,z)</math>

The exponential integral is closely related to the [[logarithmic integral function]] li(''x'') by the formula
:<math>\operatorname{li}(e^x) = \operatorname{Ei}(x)</math>
for non-zero real values of <math>x </math>.