Kate Hudson is over the moon about her baby girl

Kate Hudson is over the moon about her baby girl.

The 39-year-old mother of three “can’t stop smiling” about the latest addition to her family — newborn daughter Rani Rose, a source tells PEOPLE.

“She’s always wanted a daughter so this is really a dream come true,” the insider says.

“And her big brothers are so excited — they’ll be so protective and sweet to her always.”

Hudson welcomed her daughter alongside boyfriend Danny Fujikawa on Oct. 2. She is also mom to 7-year-old son Bingham Hawn from her relationship with ex-fiancé Matt Bellamy and 14-year-old son Ryder Russell from her marriage to ex-husband Chris Robinson.

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The proud mom shared the first photo of her daughter on Instagram Saturday, just four days after the little girl’s arrival.

“Our little rosebud,” Hudson wrote in the caption, in which her daughter who was fast asleep swaddled inside a striped blanket. In the image, the newborn wore an adorable pink hat adorned with a simple bow.

In her birth announcement, the mom of three explained the sweet meaning behind the name, Rani Rose Hudson Fujikawa.

Kate Hudson Hosts An Intimate Dinner Celebrating Fabletics
Credit: David M. Benett/Dave Benett/Getty

Kate Hudson

“We have decided to name our daughter Rani (pronounced Ronnie) after her grandfather, Ron Fujikawa. Ron was the most special man who we all miss dearly. To name her after him is an honor,” Hudson explained on Instagram. Fujikawa’s father, Ron, died in October 2012.

A source recently told PEOPLE that the couple was over the moon about their baby girl and becoming parents together.

“She and Danny are great,” the source said weeks before Hudson gave birth. “Kate is bubbly with excitement every time she talks about Danny and how they are having a family together. She is crazy about Danny.”