Influence measurement in social networks is vital to various real-world applications, such as online marketing and political campaigns. In this paper, we investigate the problem of measuring time-sensitive and topic-specific influence based on streaming texts and dynamic social networks. A user's influence can change rapidly in response to a new event and vary on different topics. For example, the political influence of Douglas Jones increased dramatically after winning the Alabama special election, and then rapidly decreased after the election week. During the same period, however, Douglas Jones' influence on sports remained low. Most existing approaches can only model the influence based on static social network structures and topic distributions. Furthermore, as popular social networking services embody many features to connect their users, multi-typed interactions make it hard to learn the roles that different interactions play when propagating information. To address these challenges, we propose a Time-sensitive and Topic-specific Influence Measurement (TTIM) method, to jointly model the streaming texts and dynamic social networks. We simulate the influence propagation process with a self-attention mechanism to learn the contributions of different interactions and track the influence dynamics with a matrix-adaptive long short-term memory. To the best of our knowledge, this is the first attempt to measure time-sensitive and topic-specific influence. Furthermore, the TTIM model can be easily adapted to supporting online learning which consumes constant training time on newly arrived data for each timestamp. We comprehensively evaluate the proposed TTIM model on five datasets from Twitter and Reddit. The experimental results demonstrate promising performance compared to the state-of-the-art social influence analysis models and the potential of TTIM in visualizing influence dynamics and topic distribution.