recursion theorem

for every computable function $\text{[math]}$ there exists an $\text{[math]}$ , called a fixed point of $\text{[math]}$ , such that $\text{[math]}$ .
[cite:;taken from @computability_soare_2016 theorem 2.2.1]

for every computable function $\text{[math]}$ , there exists $\text{[math]}$ such that $\text{[math]}$ .
[cite:;taken from @computability_soare_2016 corollary 2.2.2]