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4 Elementary FunctionsLogarithm, Exponential, Powers

§4.7 Derivatives and Differential Equations

Contents

§4.7(i) Logarithms

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=aza-1,
4.7.11 dndznza=a(a-1)(a-2)(a-n+1)za-n.

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+Be-az,

where A and B are arbitrary constants.

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