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4 Elementary FunctionsLogarithm, Exponential, Powers

§4.7 Derivatives and Differential Equations

Contents
  1. §4.7(i) Logarithms
  2. §4.7(ii) Exponentials and Powers

§4.7(i) Logarithms

4.7.1 ddzlnz=1z,
4.7.2 ddzLnz=1z,
4.7.3 dndznlnz=(1)n1(n1)!zn,
4.7.4 dndznLnz=(1)n1(n1)!zn.

For a nonvanishing analytic function f(z), the general solution of the differential equation

4.7.5 dwdz=f(z)f(z)

is

4.7.6 w(z)=Ln(f(z))+ constant.

§4.7(ii) Exponentials and Powers

4.7.7 ddzez=ez,
4.7.8 ddzeaz=aeaz,
4.7.9 ddzaz=azlna,
a0.

When az is a general power, lna is replaced by the branch of Lna used in constructing az.

4.7.10 ddzza=aza1,
4.7.11 dndznza=a(a1)(a2)(an+1)zan.

The general solution of the differential equation

4.7.12 dwdz=f(z)w

is

4.7.13 w=exp(f(z)dz)+constant.

The general solution of the differential equation

4.7.14 d2wdz2=aw,
a0,

is

4.7.15 w=Aeaz+Beaz,

where A and B are arbitrary constants.

For other differential equations see Kamke (1977, pp. 396–413).