derivation

Define "ruleeta1" for the Taylor series of is , where and .

Define "ruletheta1" for the Taylor series of , where and .

Calculate by taking the partial differentiation of with respect to . Define "rulephi2" for .

Apply for to

Substitute ruleeta1 for to get using φ derivatives as well as ψ derivatives.

The above expression is compared with , and the coefficients are obtained below.

We have obtained , and   as follows.

Let us write down the Taylor series