Is the female doctor considered responsible for the death of an infant after birth, and is expiation (kaffarah) obligatory upon her, in the event that she did not perform fetal heart rate monitoring during the first stage of labor, especially with the presence of other causes of death such as the umbilical cord wrapping around the neck and prolapsing?

We understand that you detected a weak fetal heartbeat, and you did not perform a fetal heart monitoring scan, under the pretext that you later observed its heartbeat to be normal. However, performing the scan was necessary because the weak heartbeat could recur and might not be detectable with an auscultation device.

You have fallen short in your duty, and you must seek forgiveness, repent, and resolve not to repeat such an action; for your cessation from performing what is required of you is forbidden.

However, this shortcoming of yours is not considered killing the fetus and does not necessitate expiation; because its death might have been due to your shortcoming or due to umbilical cord entanglement. As long as the cause is presumed, you are not bound by what binds a killer or an accomplice.