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Relation with other functions

Kummer's equation

is usually solved by the confluent hypergeometric functions M (a, b, z) {\displaystyle M(a,b,z)} and U (a, b, z). {\displaystyle U(a,b,z).} But when a = 0 {\displaystyle a=0} and b = 1, {\displaystyle b=1,} that is,

we have

for all z. A second solution is then given by E (− z). In fact,

with the derivative evaluated at a = 0. {\displaystyle a=0.} Another connexion with the confluent hypergeometric functions is that E is an exponential times the function U (1,1, z):

The exponential integral is closely related to the logarithmic integral function li(x) by the formula

for non-zero real values of x {\displaystyle x}.