A question about perturbation series inversion

[zetafunction]

let be m a measures (by expermients) physical quantity and m0 a 'bare' value of these physical quantity , let us suppose that we can expand

$$m= m_{0}+f(k,m_{0})+ \sum_{n} u^{n}c_{n}$$

for some finite quantities c_n and $$u=log(\Lambda)$$ with lambda a regulator

can we then invert the series above to express

$$log(\Lambda)= g( f(k,m_{0}) , m , m_{0})$$

how about if instead of logarithms of regulator there are also powers of regulator i mean quantities proportional to $$\Lambda ^{k}$$

 

[A. Neumaier]

let us suppose that we can expand

$$m= m_{0}+f(k,m_{0})+ \sum_{n} u^{n}c_{n}$$

for some finite quantities c_n and $$u=log(\Lambda)$$ with lambda a regulator

can we then invert the series above to express

$$log(\Lambda)= g( f(k,m_{0}) , m , m_{0})$$

One can formally solve every equation $$\sum_{n=1}^\infty c_nu^n=x$$
with nonzero c_1 to an equation $$\sum_{n=1}^\infty d_nx^n=u$$; simply substitute one into the other and compare coefficients to get recurrence relations.