To manually add script to the head of a WordPress site you can use the wp_head action by using one of the the following function examples in function.php If using the echo function you will need to use double quotes around scripts single quotes to keep from breaking the function.

Adding script to the head with wp_head

<?php
/**
 * add script to head with wrapping php
 */
function addtohead_cf() { 
?>
<script></script>
<?php 
}
add_action('wp_head', 'addtohead_cf');
/**
 * add script to head using echo
 */
function addtohead_cf() {
echo "<script></script>";
}
add_action( 'wp_head', 'addtohead_cf' );