51.6. LIOUVILLE’S THEOREM 1631

Note how radically different this is from the theory of functions of a real variable.Consider, for example the function

f (x)≡{

x2 sin( 1

x

)if x ̸= 0

0 if x = 0

which has a derivative for all x ∈ R and for which 0 is a limit point of the set, Z, eventhough f is not identically equal to zero.

Here is a very important application called Euler’s formula. Recall that

ez ≡ ex (cos(y)+ isin(y)) (51.5.19)

Is it also true that ez = ∑∞k=0

zk

k! ?

Theorem 51.5.4 (Euler’s Formula) Let z = x+ iy. Then

ez =∞

∑k=0

zk

k!.

Proof: It was already observed that ez given by 51.5.19 is analytic. So is exp(z) ≡∑

∞k=0

zk

k! . In fact the power series converges for all z ∈ C. Furthermore the two functions,ez and exp(z) agree on the real line which is a set which contains a limit point. Therefore,they agree for all values of z ∈ C.

This formula shows the famous two identities,

eiπ =−1 and e2πi = 1.

51.6 Liouville’s TheoremThe following theorem pertains to functions which are analytic on all of C, “entire” func-tions.

Definition 51.6.1 A function, f :C→ C or more generally, f :C→ X is entire means it isanalytic on C.

Theorem 51.6.2 (Liouville’s theorem) If f is a bounded entire function having values inX , then f is a constant.

Proof: Since f is entire, pick any z ∈ C and write

f ′ (z) =1

2πi

∫γR

f (w)

(w− z)2 dw

where γR (t) = z+Reit for t ∈ [0,2π] . Therefore,

∣∣∣∣ f ′ (z)∣∣∣∣≤C1R

51.6. LIOUVILLE’S THEOREM 1631Note how radically different this is from the theory of functions of a real variable.Consider, for example the function_ x* sin (4) ifx 40faye Oifx=0which has a derivative for all x € R and for which 0 is a limit point of the set, Z, eventhough f is not identically equal to zero.Here is a very important application called Euler’s formula. Recall thate& =e" (cos(y) + isin (y)) (51.5.19)Is it also true that e* = V4 a)Theorem 51.5.4 (Euler’s Formula) Let z= x-+iy. Thenco okZze= y —,Lok!Proof: It was already observed that e* given by 51.5.19 is analytic. So is exp(z) =Yeo z In fact the power series converges for all z € C. Furthermore the two functions,é and exp (z) agree on the real line which is a set which contains a limit point. Therefore,they agree for all values of z € C.This formula shows the famous two identities,imel =—lande?™ =1.51.6 Liouville’s TheoremThe following theorem pertains to functions which are analytic on all of C, “entire” func-tions.Definition 51.6.1 A function, f :C — C or more generally, f : C — X is entire means it isanalytic on C.Theorem 51.6.2 (Liouville’s theorem) If f is a bounded entire function having values inX, then f is a constant.Proof: Since f is entire, pick any z € C and writeKo= pf Lawni (w—z)?where Yp (t) =z+Re" fort € [0,27]. Therefore,I all<cR