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While the airline decided not to identify the pilot, saying only that the employee has years experience on the job, passengers posting on social media identified the pilot as Tammie Jo Shults.
She was calm, cool and collected, they said, safely landing the plane with 144 passengers and five crew members on board despite the loss of one of its engines and a shattered window in the cabin.
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Passenger Amanda Bourman posted, "The pilot, Tammy Jo, was amazing!"
Another wrote, "(She) came back to speak to each of us personally. This is a true American hero. A huge thank you for her knowledge, guidance, and bravery."
KGO's sister station WPVI-TV has learned that Shults is from New Mexico and graduated from MidAmerica Nazarene University. She was one of the Navy's first female fighter pilots.
"She has nerves of steel. That lady, I applaud her," said Alfred Tumlinson, of Corpus Christi, Texas. "I'm going to send her a Christmas card, I'm going to tell you that, with a gift certificate for getting me on the ground. She was awesome."
While he did not single out the pilot, Southwest CEO Gary Kelly praised all the crew members, saying, "They did their jobs superbly today."
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He called the incident a tragic loss while praising the crew for landing the aircraft without further loss of life.
One passenger, Jennifer Riordan, a Wells Fargo bank executive and mother of two from Albuquerque, New Mexico, died after being partially pulled out of the plane after a cabin window shattered.
The National Transportation Safety Board is leading the investigation into what caused the engine to fail.
"Obviously very grateful there were no other injuries other than one fatality. By all reports, I thought the crew acted magnificently," said Kelly.
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Passenger Kathy Farnan said, "They were also taking care of everybody with running around with oxygen, making sure everybody had enough oxygen."
On Twitter, Kristopher Johnson said, "These are the heroes of SW 1380. We lost an engine mid-flight and they guided us back to Philly."
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The airline said it would make sure the flight crew is feeling fit for duty before they head back in the air.
The Associated Press contributed to this report.
Click here for a look at recent stories and videos about emergency landings here in the Bay Area and across the country.