1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 275

the exponential also has a meaning when the exponent is any complex

number, the following formulas, generalizing (5.19-3), are used to define

cos z and sin z in the complex domain:

eiz + e-iz

cosz= ----

2

eiz _ e-iz

sinz= ----

2i

(5.19-4)

From the formulas above, the same circular or trigonometric identities

known in the real domain are derived easily. Thus we have

1. cos( -z) = cos z, sin( -z) = -sin z


2. cos(z + 2k7r) = cosz, sin(z + 2k7r) = sinz


3. cos^2 z + sin^2 z = 1


4. cos( z1 + z2) = cos z 1 cos z 2 - sin z 1 sin z 2


5. sin( z1 + z2) = sin z1 cos z 2 + cos z 1 sin z 2


6. cos2z = cos^2 z - sin^2 z = 2cos^2 z -1=1-2sin^2 z


7. sin 2z = 2 sin z cos z
   etc.

Also, we note that from (5.19-4) it follows that

eiz = cosz + isinz (5.19-5)

which generalizes (5.19-1). The remaining circular functions are defined in
terms of sine and/ or cosine as usual. Thus

sinz 1 eiz - e-iz. e2iz -1

tan z = --= - = -i --,,---

cos z i eiz + e-iz e^2 iz + 1

1 cos~ 1

cotz =

tanz

1

= secz = cscz =

sinz cosz sinz

The well-known identities

tan^2 z + 1 = sec^2 z, cot^2 z + 1 = csc^2 z

still hold good. It will be shown later that the validity of all the circular

(and hyperbolic) identities in the complex domain is a consequence of the

so-called identity principle for analytic functions. It is to be understood

that tanz and secz are not defined for z =^1 / 2 (2k + l)Tr, while cotz and
cscz are not defined for z = kTr (k any integer).
As to the hyperbolic functions, they are defined by the formulas
corresponding to those of the real case, namely,

coshz =

ez + e-z

sinhz =

ez - e-z

2 2

sinhz

cothz =

1

tanhz = (5.19-6)

coshz ' tanhz