Rodgers took over at Anfield last week after admitting the opportunity was too good to turn down following a second approach from the Merseysiders.
The Northern Irishman admitted he received three opportunities to leave the Liberty Stadium in the past, but the offer from Liverpool was on another level.
"The chance to manage one of the world's great footballing institutions doesn't come around very often in a manager's life and, after two approaches, I have decided to take up the professional challenge," he wrote in his letter, published by the South Wales Echo.
"Before doing so I wanted to leave you my wholehearted thanks and gratitude for a truly memorable period of my life.
"My story at Swansea City began nearly two years ago and my mission above anything else was very clear: to inspire the city.
"My aim for a legacy was simply to have made Swansea City Football Club an attacking and creative team who were enjoyable to watch, and to make the people of Swansea proud of their great club, city and nation.
"In the years ahead, you, the most important people at the club, will decide if I achieved those dreams, but the journey for me has been one I will never forget.
"We created history together to become the first Welsh club in the Premier League, and defied all the odds at the same time, and there is no doubt the club will continue to flourish in the coming years."
Rodgers continued: "The players at Swansea I will thank individually, but to them I hope I have helped you become better players and people because you have certainly done that for me.
"And, finally, to the Jack Army. I'm proud to have been your manager and feel truly blessed for the support you have given me on and off the field.
"I leave knowing I couldn't have given any more to help your great club and succeed.
"Swansea will live in my heart for the rest of my life and one day I would love to return to this great club and city.
"Until that day comes, I wish everyone at Swansea lots of success in the next chapter of your amazing story."
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