|eᶻ| < 1 there, but f is not constant. Liouville needs the whole plane.
Counterexample bank
Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.
AllODEPDEabeliananalyticityanovabairecalculus of variationscanonical formscauchy-riemanncompactnesscompletenesscomplexconformalconnectednesscontinuitycontour integrationconvergencedifferentiationdistributionsfieldsfixed pointgaloisgreengroupsharmonicinjectivityintegral equationsintegrationlaurentlimsuplinear algebraliouvillelpmarkovmeasuremechanicsmetric spacesmultivariablemultivariatenumericalp-groupspermutationspolynomialspower seriesprobabilityquadratic formsreal analysisregressionresiduesringssamplingseparabilityseparationsequencesseriessingularitiesstabilitystatisticssylowtestingtopologyuniform convergencewronskianzeros
#1 · Complex Analysis, Algebra & Topology › Cauchy Theory
“Bounded real part ⇒ entire function is constant — fails if only |f| is bounded on a half-plane” — false
Counterexample locked — unlock with Notes + PYQ
complexliouville
#2 · Complex Analysis, Algebra & Topology › Cauchy Theory
“A bounded holomorphic function on an unbounded domain is constant” — false
Counterexample: f(z) = eᶻ on {Re z < 0}
complexliouville
#3 · Complex Analysis, Algebra & Topology › Cauchy Theory
“An entire function omitting one value is constant” — false
Counterexample locked — unlock with Notes + PYQ
complexliouville