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1.10 Complex Methods 113

v=0? Sometimes notation is compressed and, instead of the last line, we

write

f(x)=

∫∞

−∞

f(t)δ(t−x)dt.

Althoughδis not, strictly speaking, a function, it is calledDirac’s delta func-

tion.

1.10 Complex Methods

Fourier series

Suppose that a functionf(x)equals its Fourier series

f(x)=a 0 +

∑∞

n= 1

ancos(nx)+bnsin(nx).

(We use period 2πfor simplicity only.) A famousformula of Euler states that

eiθ=cos(θ )+isin(θ ), wherei^2 =− 1.

Some simple algebra then gives the exponential definitions of the sine and
cosine:

cos(θ )=^1

2

(

eiθ+e−iθ

)

, sin(θ )=^1

2 i

(

eiθ−e−iθ

)

.

By substituting the exponential forms into the Fourier series offwe arrive at
the alternate form

f(x)=a 0 +^1

2

∑∞

n= 1

an

(

einx+e−inx

)

−ibn

(

einx−e−inx

)

=a 0 +^1

2

∑∞

n= 1

(an−ibn)einx+(an+ibn)e−inx.

We a r e n o w l e d t o d e fi n ecomplex Fourier coefficientsforf:

c 0 =a 0 , cn=^1

2

(an−ibn), c−n=^1

2

(an+ibn), n= 1 , 2 , 3 ,....

In terms of these two coefficients, we have

f(x)=c 0 +

∑∞

n= 1

(

cneinx+c−ne−inx

)

=

∑∞

−∞

cneinx. (1)