JAKARTA, Indonesia — President Obama, renewing his call for better relations between the United States and the Muslim world, used a long-awaited homecoming trip to this island nation to make a symbolic visit on Wednesday morning to the largest mosque in southeast Asia — even as he declared that “much more work needs to be done” to fulfill the promise he made 17 months ago in Cairo of a “new beginning.”

Indonesia is the world’s largest majority Muslim nation, and Mr. Obama, on a 10-day, four-country trip through Asia, used his brief stay here to hold it up as an example of diversity, tolerance and democracy.

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