Most purchases made on an iPhone go off without a hitch, but occasionally a download will not advance past the processing step. In most cases, this means you simply need to connect to a nearby Wi-Fi network. When the screen freezes altogether, however, you need to reboot the hard drive on your iPhone.
If an iTunes purchase gets stuck on the processing screen while it is downloading, it might need wireless Internet connection. While 3G connection will typically suffice for song purchases and most applications, larger downloads need Wi-Fi connection to complete. Videos and large applications are the most likely to fall into the category of downloads that require Wi-Fi. To turn on Wi-Fi when the download is stuck, go to the "Settings" application and tap "Wi-Fi." Toggle the "Wi-Fi" setting to "On" and then return to iTunes to watch your download complete.
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If you begin to download a video on your iPhone and it freezes but you don't have access to Wi-Fi, you can always cancel the download. Just open the iTunes screen showing the download and swipe your finger across the download from left to right. Tap the red "Delete" button that appears to start the download. Initiate the download again in the future when you have access to Wi-Fi; iTunes won't charge you twice for the same download.
If your iPhone is completely stuck on the processing screen, as in it won't navigate to another screen, try starting the device. To restart an iPhone, hold down the "Sleep/Wake" button on the top of your phone until the red slider appears along the top of the screen. Move the arrow from the left side of the slider to the right, and the iPhone will power down. Press the "Sleep/Wake" again to power the iPhone on and try the download again.
Sometimes the iPhone gets stuck to the point where it will not restart. You can force your iPhone to reboot by holding down the "Sleep/Wake" button and "Home" button at the same time. Once a white Apple logo appears on the screen, the phone is rebooting.