Page 39 - Calculus with Complex Numbers

as already observed above.
To see where the M aclaurin formula for the rlth coeflicient comes from observe
that if
flx) = aè + a3x + 52.12 + JIS-'rS + . . .

then putting .x = O gives f(Q) = ao.
Differentiating term by term we get
f'lx) = Jl + 252.1 + 353.r2 + . . . ,

which on substituting .x = 0 gives f' (0) = tz1 .
Differentiating again we get
y// (.x) = 1az + 6..x2 + . . . ,

which on substituting .x = 0 gives fn (0) = ztzz, and hence az = fM (0)/21
Similarly, differentiating n times and putting .x = 0 we get fln) (0) = n ltzn,
and hence an = f (rl) (Q)(n ! as required.
M aclaurin was concerned with real variables only but his exparlsion remains
valid for complex variables also. W e list below some examples of M aclaurin
expansions in the complex context.
z 2 z 3
ez = 1 + z + - + - + . . .
2 ! 3 !
z z 5
3
sin z = z - - + - . . .
3! V
2 4
. E .E
cos z = 1 - - + - - . . .
2 ! 4 !
3 5
.E .E
sinh z = z + - + + . . .
3! V
z 2 z 4
cosh z = 1 + - + - + . . .
2 ! 4 !
(1 + a (a - 1) z
zllf = 1 + az + 2 ! z + . . .

(1 z I < 1)

( Iz I< 1) .