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Here's something for you folks. It was possible to file for a Hawaiian birth certificate, days after the birth; even if there was no medical professional or midwife present at the birth. It would basically involve a signed statement by the parent that "so and so child" was born in Hawaii on "such and such day", at 'such and such time', etc.. He are the provisions that allowed for that at the time :

"Exactly what kinds of birth records does Hawaii provide?
Posted by Mijgreb on Thursday, February 05, 2009 12:00:00 AM
Generally, folks don't know that Hawaii law, even in 1961, provided for multiple kinds of birth records, most of which are not what people think of when they think of birth certificates. The following is a description of those, including certificates for people not born in Hawaii. Go figure!
1. In the State of Hawaii, back in 1961, there were three different birth certificates that were obtainable:
a. If the birth was attended by a physician or mid wife, the attending medical professional was required to certify to the Department of Health the facts of the birth date, location, parents’ identities and other information. (See Section 57-8 & 9 of the Territorial Public Health Statistics Act in the 1955 Revised Laws of Hawaii which was in effect in 1961).
b. In 1961, if a person was born in Hawaii but not attended by a physician or mid wife, then, up to the first birthday of the child, an adult could, upon testimony, file a “Delayed Certificate”, which required endorsement on the Delayed Certificate of a summary statement of the evidence submitted in support of the acceptance for delayed filing, which evidence must be kept in a special permanent file.
The statute provided that the probative value of the Delayed Certificate must be determined by the judicial or administrative body or official before whom the certificate is offered as evidence. (See Section 57-18, 19 & 20 of the Territorial Public Health Statistics Act in the 1955 Revised Laws of Hawaii which was in effect in 1961)."
- http://mijgreb.blogtownhall.com/

Now if Obama was actually born in Kapolei Medical Center then there should be an actual hospital certificate with the names and signatures of the medical staff who were working there. Even if Ann Dunham had acccidently lost the certificate, she could go back and get some sort of duplicate form the hospital.But, if there no hospital document, then you obviously have to ask why there is no certificate or document from a hosptial. Again, there absolutely should be if he was born at K.M.C..

If you ordered a Certification of Live Birth from the DoH, then it would simply provide the basic information from the parent's testimony that was submitted to the DoH after the child's birth.