The first volume ended in March 1954, when Sinatra, the down-on-his-luck crown prince of Hoboken, New Jersey, received the best supporting actor Oscar for his role as the similarly down-trodden Private Angelo Maggio in From Here to Eternity; '... he cleverly thanked everybody by thanking nobody,' wrote Kaplan. With that, Sinatra ducked his own celebration party and took a walk through Beverly Hills: 'Just me and Oscar!'

Volume II picks up the story nine days later, when Sinatra types a note, in lower case and the third person, to a friend: 'dear lew – my paisan mr sinatra is still on cloud nine and the bum refuses to come down ...'

In fact, Sinatra actively defied the remark by talent agent Irving 'Swifty' Lazar, who notoriously declared, 'He's a dead man. Even Jesus couldn't get resurrected in this town.' The moment he clutched his Oscar ignited what Kaplan describes as 'the greatest comeback in show-business history'. It is this that precipitated a career whose trajectory seldom wavered over the next forty-four years. It made Frank Sinatra a prolific and celebrated performer and recording artist; it brought him astonishing wealth, friendships celebrated and dubious, and a legion of beautiful women. It might be easier to list those women he didn't take to bed – Eleanor Roosevelt, one presumes, resisted that Ol' Blue Eyes' charm, and Jacqueline Kennedy detested Sinatra and his dubious connections to organised crime (though she dated him briefly after Aristotle Onassis's death).

Kaplan does well – in fact, one suspects, far better and more scrupulously than any Sinatra biographer before or since – in drawing together the strands of a story that must be as complete as possible in order properly to survey its subject. Sinatra, for all his exacting and stupendous vocal and creative genius, for all his generosity and kindness to friends, was a disturbed, often vengeful and violent, human being. One of his greatest songs, 'I've got you under my skin', fails to reveal that Frank's own skin was tissue-thin; scratch it, and he bled easily. In the index, under Sinatra's own entry, you find the neuroses and foibles nestled in with the more appreciable elements. Thus: 'acting skill of', 'aggression', 'apologising as difficult for', 'arrogance of', 'humiliation of', 'impatience of', 'insecurities of'; 'vindictiveness of', 'violent temper of' etc.

'Sinatra: The Chairman shows the darker side: no longer the boy singer, but someone more worldly-worn'

In 1956, at one of the twenty-one recording sessions he would make that year for Capitol, Sinatra said to his masterly arranger, Nelson Riddle, 'I want a long crescendo.' He got what he asked for – not just for the song in question, but for his own fame and reputation. By now, earning an estimated $US1 million a year (more than $US870 million in today's money), Sinatra was as much a corporate figure as a performer. By the end of the decade he would sever his relationship with Capitol to start his own label, Reprise. He could pick and choose his film roles, occasionally shaping them to his own personality (that memorable 'ring-a-ding-ding' inserted into one of his songs in Can-Can, 1960, was not Cole Porter's idea).

All the while, Sinatra's flipside, epitomised by his incursions into Las Vegas casinos and his friendship with Mob boss Sam Giancana, was on the rise – as was, more temporarily, Sinatra's association with John F. Kennedy, which was consigned to the back of the freezer once Kennedy became president.

Frank Sinatra at a dinner at the White House in honour of Giulio Andreotti, President of the Council of Ministers of the Italian Republic (left), and Richard Nixon, 1973 (photograph by Jack E. Kightlinger, White House photographer, via Wikimedia Commons)

Then there is love – a word Sinatra could sing with such ease and style that one almost believed he believed in it as a singular experience. Well, not quite. When he introduced his new wife, Mia Farrow, to a Las Vegas audience in 1966, he said, 'Yeah, I sure got married. Well, you see, I had to – I finally found a broad I can cheat on.' Not that much later on, Sinatra filed for divorce.

The Sinatra story could effectively have concluded in 1971, when Ol' Blue Eyes officially retired. Two years later, Sinatra was back, in a different mode: the saloon singer performed mainly to arena-size audiences; he starred in just two films, one of them for television; and even his politics had swung the other way, with his support of Republican presidential candidate Richard Nixon and, later, Ronald Reagan. His fourth, and final, marriage, to Barbara Marx, was not exactly idyllic. As Sinatra's younger daughter, Tina, wrote: 'Barbara could extract the deep, dark anger out of Dad ... She had a street fighter's résumé, and Dad was a withering counter-puncher. They hit each other below the belt and didn't stop when the bell rang.' The bell finally tolled for Frank Sinatra on 14 May 1998.

By the end – it is worth mentioning that the Sinatra comeback and decline is covered in fewer than 40 pages – we know an awful lot about an awful man who was a formidable friend and deadly enemy and who happened to be one of the most significant and influential musicians of the twentieth century. Is all of it worth knowing? Do we need guest list after guest list? Even in the interests of complete research, is it worth recycling some of the grubbier information (for example, from Kitty Kelley's discredited 1983 biography)?

The first volume was more tightly written, more dramatically propulsive, more exciting to read. The second, however, even with more ground to cover, is flabbier in construction and in need of editing. Even long crescendos shouldn't go on forever. Still, Sinatra admirers, with a bit of perseverance, will love it.