Federated learning (FL) has recently emerged as a compelling machine learning paradigm, prioritizing the protection of privacy for training data. The increasing demand to address issues such as ``the right to be forgotten'' and combat data poisoning attacks highlights the importance of techniques, known as \textit{unlearning}, which facilitate the removal of specific training data from trained FL models. Despite numerous unlearning methods proposed for centralized learning, they often prove inapplicable to FL due to fundamental differences in the operation of the two learning paradigms. Consequently, unlearning in FL remains in its early stages, presenting several challenges. Many existing unlearning solutions in FL require a costly retraining process, which can be burdensome for clients. Moreover, these methods are primarily validated through experiments, lacking theoretical assurances. In this study, we introduce Fast-FedUL, a tailored unlearning method for FL, which eliminates the need for retraining entirely. Through meticulous analysis of the target client's influence on the global model in each round, we develop an algorithm to systematically remove the impact of the target client from the trained model. In addition to presenting empirical findings, we offer a theoretical analysis delineating the upper bound of our unlearned model and the exact retrained model (the one obtained through retraining using untargeted clients). Experimental results with backdoor attack scenarios indicate that Fast-FedUL effectively removes almost all traces of the target client, while retaining the knowledge of untargeted clients (obtaining a high accuracy of up to 98\% on the main task). Significantly, Fast-FedUL attains the lowest time complexity, providing a speed that is 1000 times faster than retraining. Our source code is publicly available at \url{https://github.com/thanhtrunghuynh93/fastFedUL}.