Fix functions-compat

Author: hsubox76

packages-exp/functions-compat/src/register.ts

@@ -29,6 +29,8 @@ import {
 } from '@firebase/component';
 import { Functions as FunctionsServiceExp } from '@firebase/functions-exp';
 
+const DEFAULT_REGION = 'us-central1';
+
 declare module '@firebase/component' {
   interface NameServiceMapping {
     'app-compat': FirebaseApp;
@@ -46,7 +48,7 @@ const factory: InstanceFactory<'functions-compat'> = (
   const functionsServiceExp = container
     .getProvider('functions-exp')
     .getImmediate({
-      identifier: regionOrCustomDomain
+      identifier: regionOrCustomDomain ?? DEFAULT_REGION
     });
 
   return new FunctionsService(app, functionsServiceExp);

packages-exp/functions-exp/src/config.ts

@@ -25,8 +25,6 @@ import {
 } from '@firebase/component';
 import { FUNCTIONS_TYPE } from './constants';
 
-export const DEFAULT_REGION = 'us-central1';
-
 export function registerFunctions(fetchImpl: typeof fetch): void {
   const factory: InstanceFactory<'functions'> = (
     container: ComponentContainer,