§1. Convergent sequences and series
81
In practice, all the functions that one meets in classical analysis are representable as power series or, if necessary, by series with a further finite number
of terms of negative degree like
or by series of this type where the variable is a fractional power of x, as in
the relation
which Newton deduced from the binomial series for s = 1/2. This fact led
the mathematicians to study systematically from the XIXth century - there
was an attempt by Lagrange a little earlier which came to nothing because
he restricted himself to functions of a real variable - the analytic functions
of a complex variable, which one can define as follows. Consider a function J
with complex values defined on an open subset G of C, Le. such that, for all
a E G, the set G contains an open disc dCa, z) < r with centre a and radius
r > 0 (depending on a). The function J is called analytic if, for all a E G,
there exists a power series
Co(a) + cl(a)(z - a) + c2(a)(z - a)2 + ... = L en(a)(z - a)n
whose coefficients depend on a and which (i) converges for Iz - al sufficiently
small, (ii) has sum J(z) on a neighbourhood of a, whence necessarily coCa) =
f(a). Take for example the function J(z) = liz, defined on the open set
z =f. O. For a =f. 0, one can write
~=
1
=~
1
=~(a_z)nlan+1
z
a-(a-z)
a1-(a-z)la
~
nEN
on condition that I(a - z)lal < 1, i.e. Iz - al < lali the function liz is thus
represented by the power series in z - a that we have just written, in the
largest disc with centre a not containing - which is normal - the point z = o.
Here en(a) = (_l)nla n +1.
Cauchy (1789-1857) was the first to observe, having consecrated thirty
years of work to them before seeing it clearly, that these functions possess
extraordinary properties which nearly all the mathematicians of the XIXth
century, and a good subset of their successors, have put to work from one
time to an other, and have in particular generalised to functions of several
complex variables.
7 - The marvels of the harmonic series
Let us return to the elementary theory of convergent sequences. Remember
that a sequence (vn ) is a subsequence of a sequence (un) if there exists a
81
In practice, all the functions that one meets in classical analysis are representable as power series or, if necessary, by series with a further finite number
of terms of negative degree like
or by series of this type where the variable is a fractional power of x, as in
the relation
which Newton deduced from the binomial series for s = 1/2. This fact led
the mathematicians to study systematically from the XIXth century - there
was an attempt by Lagrange a little earlier which came to nothing because
he restricted himself to functions of a real variable - the analytic functions
of a complex variable, which one can define as follows. Consider a function J
with complex values defined on an open subset G of C, Le. such that, for all
a E G, the set G contains an open disc dCa, z) < r with centre a and radius
r > 0 (depending on a). The function J is called analytic if, for all a E G,
there exists a power series
Co(a) + cl(a)(z - a) + c2(a)(z - a)2 + ... = L en(a)(z - a)n
whose coefficients depend on a and which (i) converges for Iz - al sufficiently
small, (ii) has sum J(z) on a neighbourhood of a, whence necessarily coCa) =
f(a). Take for example the function J(z) = liz, defined on the open set
z =f. O. For a =f. 0, one can write
~=
1
=~
1
=~(a_z)nlan+1
z
a-(a-z)
a1-(a-z)la
~
nEN
on condition that I(a - z)lal < 1, i.e. Iz - al < lali the function liz is thus
represented by the power series in z - a that we have just written, in the
largest disc with centre a not containing - which is normal - the point z = o.
Here en(a) = (_l)nla n +1.
Cauchy (1789-1857) was the first to observe, having consecrated thirty
years of work to them before seeing it clearly, that these functions possess
extraordinary properties which nearly all the mathematicians of the XIXth
century, and a good subset of their successors, have put to work from one
time to an other, and have in particular generalised to functions of several
complex variables.
7 - The marvels of the harmonic series
Let us return to the elementary theory of convergent sequences. Remember
that a sequence (vn ) is a subsequence of a sequence (un) if there exists a