Rik Mayall suffered an "acute cardiac event" after returning from a morning run, his widow has said.
Barbara Mayall said her family had received "thousands and thousands of messages of condolence" from around the world.
She said: "We always knew that Rik was well loved but we are all overwhelmed by so many joining us in our grief.
"We now know that our darling Rik suffered an 'acute cardiac event' at our home around midday on June 9th.
"He had just returned from his usual run and many people had seen him that morning."
Mrs Mayall thanked the press "for their discretion" and "all the wonderful coverage", adding: "It would have made him very happy."
She added: "I am sure that you all know Rik's response would be something along the lines of.....' well thanks very much all of you.............now f*** off!!!'"
A post-mortem into the comedian's death was inconclusive, West London Coroner's Court has said.
More tests will now be carried out in an attempt to determine how the 56-year-old star died.
Rik, who shot to fame playing poetry-writing anarchist Rick in The Young Ones, was found dead at his London home on June 9.
His career, which also included roles such as Lord Flashheart in Blackadder and the conniving Conservative MP Alan B'Stard in The New Statesman, stretched back to the start of the alternative comedy scene.
Tributes have flooded in for Rik since his death.
His daughter Bonnie said on Facebook: "My dad was loved not only by my family, but by many many others.
"We will never forget him and neither will the world.
"RIP to the man, the myth, the legend - my wonderful, generous, foul mouthed and hysterical father. My idol now and forever.
"We love you daddy."
His Comic Strip Presents... colleague Peter Richardson, whose son was one of the last people to see the actor alive, said he was happy and healthy in the hours before his death.
Peter, who directed Rik in a series of TV shows, told BBC Radio 4's Today programme that his son saw him around half an hour before he died.
He said: "He was happily chatting away and it was very quick, and we still don't quite know what happened, but it was a seizure of some sort."
Rik, who leaves wife Barbara and three children, Rosie, Sidney and Bonnie, survived an almost-fatal quad bike accident in 1998 which left him in a coma for several days.
Close friend and long-time collaborator Adrian Edmondson said he felt privileged to have shared good times with him at Manchester University, where the pair studied.
He said: "There were times when Rik and I were writing together when we almost died laughing. They were some of the most carefree stupid days I ever had, and I feel privileged to have shared them with him. And now he's died for real. Without me. Selfish b*****d."
Stephen Fry, who also starred in Blackadder, said on Twitter: "Simply distraught to hear of the death of Rik Mayall. An authentic comedy genius and a prince among men."
Ben Elton, who was also a university friend of Mayall, said: "I owe him so much. He changed my life utterly when he asked me to co-write The Young Ones with him and he was with me on the day I met my wife. He always made me cry with laughter, now he's just made me cry."
Another less orthodox tribute - in the form of a makeshift blue plaque - was put up in Hammersmith, west London, reading: "Rik Mayall 1958-2014 Punched his friend in the balls on a bench near this spot."
A Scotland Yard spokesman said officers were called by London Ambulance Service to a house in Barnes, south west London, where "a man, aged in his 50s, was pronounced dead at the scene".
He added that the death was not believed to be suspicious.
Rik was born in Harlow, Essex, to drama teacher parents and launched his comedy career on stage in a duo, the Dangerous Brothers, with Ade Edmondson.