The Splunk server name of the member can typically be determined by the serverName attribute in the server.conf file, which is not explicitly shown in the provided snippet. However, based on the provided configuration snippet, we can infer that this search head cluster member is configured to communicate with a cluster master (master_uri) located at node1 and a management node (mgmt_uri) located at node3. The serverName is not the same as the master_uri or mgmt_uri; these URIs indicate the location of the master and management nodes that this member interacts with.

Since the serverName is not provided in the snippet, one would typically look for a setting under the [general] stanza in server.conf. However, given the options and the common naming conventions in a Splunk environment, node3 would be a reasonable guess for the server name of this member, since it is indicated as the management URI within the [shclustering] stanza, which suggests it might be the name or address of the server in question.

For accurate identification, you would need to access the full server.conf file or the Splunk Web on the search head cluster member and look under Settings > Server settings > General settings to find the actual serverName. Reference for these details would be found in the Splunk documentation regarding the configuration files, particularly server.conf.