Edit in JSFiddle

(function(angular) {
  'use strict';

  var app = angular.module("app", []);
  app.factory('$exceptionHandler', ['$log', '$injector', '$svcLog', svcExceptionHandler]);
  app.factory('$svcLogHttp', ['$http', svcLogHttp]);
  app.factory('$svcLog', [svcLog]);

  app.controller("ctrlHome", ["$scope",
    function($scope) {
      console.log('controller init');

      $scope.createError = function() {
        invalid.data = 'this is an error on the controller';
      }

      $scope.$on('app-error', function(event, args) {
        $scope.error = args.message;
        $scope.stack = args.stack;
      });

    }
  ]);

  //using $http service. this creates a circular reference
  function svcLogHttp($http) {

    var svc = {
      add: add
    }

    function add(exception) {
      //simulate sending the error here		
      $http.post('/api/log', exception).then(function(resp) {
        console.log();
      }, function(err) {
        console.log(err);
      });
    }
    return svc;
  }

  //just using jquery no circular reference
  function svcLog() {

    var svc = {
      add: add
    }

    function add(exception) {
      //simulate sending the error here
      var data = angular.toJson(exception);
      console.log('Sending to the server - ' + data);
      $.ajax({
        type: "POST",
        url: "/api/log",
        contentType: "application/json",
        data: data
      });
    }

    return svc;
  }

  //this service uses either the injected $svcLog or inject $svcLogHttp to avoid circular reference
  function svcExceptionHandler($log, $injector, $svcLog, $rootScope) {
    var $svc = $svcLog;
    var $rootScope = null;
    var handler = function(exception, cause) {
      var ex = {
        message: exception.message,
        stack: exception.stack
      };
      //log to console and allow the app to continue
      $log.error.apply($log, arguments);

      try {
        if (!$svc) {
          $svc = $injector.get('$svcLogHttp');
        }

        if (!$rootScope) {
          $rootScope = $injector.get('$rootScope');
        }
        //send the error
        $rootScope.$broadcast('app-error', ex);
        $svc.add(ex);
      } catch (err) {
        $log.log(err);
      }
    };

    return handler;
  }

}(angular));

<div ng-app="app">
  <div ng-controller="ctrlHome as ctrl">
    <div class="body-content panel-body text-center">
      <h2 class="text-info">AngularJS Server Side Exception Logging by ozkary.com</h2>
      <div class="row">
        <div class="col-sm-12">
          <div class="">
            <input class="btn btn-primary btn-large" type="button" ng-click="createError()" value="Click to raise error" />
          </div>
        </div>
      </div>
      <div class="row">
        <div class="col-sm-12">
          <h3>Message</h3>
          <div ng-bind="error" class="bg-warning container" style="padding:15px;">
          </div>
          <h2>Stack</h2>
          <div ng-bind="stack" class="bg-warning container" style="padding:15px;">
          </div>
        </div>
      </div>
    </div>
  </div>
</div>