[SalesForce] How to refresh Static Resource JS file in Lightning Component

You should be able to do this right inside of your controller code. However, you need to make sure that the call to this function doesn't happen until the script is loaded. This is done by specifying a controller action to be called in the afterScriptsLoaded parameter of <ltng:require>.

In your component:

<ltng:require scripts="/resource/geoCode" afterScriptsLoaded="{!c.scriptsLoaded}"/>

In your controller:

({
  scriptsLoaded : function(component, event, helper) {
    callMe();
  }
})

See this for the official docs: Using External JavaScript Libraries with Lightning Components

you could attach your utility methods to the global window object(SecureWindow) and use it in your controller.

Static Resource: TestScript

(function(w){
    "use strict"; //Optional because LC on LockerService active runs in strict mode

    var utilMethods = {
        "method1":method1,
        "method2":method2
    };

    function method1(){
        console.log("method1 called");
    } 

    function method2(){
        console.log("method2 called");
    }

    w.myUtil = utilMethods;

})(window);

Component:

<ltng:require scripts="{!$Resource.TestScript}" afterScriptsLoaded="{!c.afterScriptsLoaded}" />
<ui:button label="Do job" press="{!c.doJob}"/>

Controller.js:

doJob : function(cmp){
     myUtil.method1(); // Print method1 called
     myUtil.method2(); // Print method2 called
}