How can I get a list of all functions stored in the database of a particular schema in PostgreSQL?

\df <schema>.*

in psql gives the necessary information.

To see the query that's used internally connect to a database with psql and supply an extra "-E" (or "--echo-hidden") option and then execute the above command.

After some searching, I was able to find the information_schema.routines table and the information_schema.parameters tables. Using those, one can construct a query for this purpose. LEFT JOIN, instead of JOIN, is necessary to retrieve functions without parameters.

SELECT routines.routine_name, parameters.data_type, parameters.ordinal_position
FROM information_schema.routines
    LEFT JOIN information_schema.parameters ON routines.specific_name=parameters.specific_name
WHERE routines.specific_schema='my_specified_schema_name'
ORDER BY routines.routine_name, parameters.ordinal_position;

If any one is interested here is what query is executed by psql on postgres 9.1:

SELECT n.nspname as "Schema",
  p.proname as "Name",
  pg_catalog.pg_get_function_result(p.oid) as "Result data type",
  pg_catalog.pg_get_function_arguments(p.oid) as "Argument data types",
 CASE
  WHEN p.proisagg THEN 'agg'
  WHEN p.proiswindow THEN 'window'
  WHEN p.prorettype = 'pg_catalog.trigger'::pg_catalog.regtype THEN 'trigger'
  ELSE 'normal'
 END as "Type"
FROM pg_catalog.pg_proc p
     LEFT JOIN pg_catalog.pg_namespace n ON n.oid = p.pronamespace
WHERE pg_catalog.pg_function_is_visible(p.oid)
      AND n.nspname <> 'pg_catalog'
      AND n.nspname <> 'information_schema'
ORDER BY 1, 2, 4;

You can get what psql runs for a backslash command by running psql with the -E flag.

There's a handy function, oidvectortypes, that makes this a lot easier.

SELECT format('%I.%I(%s)', ns.nspname, p.proname, oidvectortypes(p.proargtypes)) 
FROM pg_proc p INNER JOIN pg_namespace ns ON (p.pronamespace = ns.oid)
WHERE ns.nspname = 'my_namespace';

Credit to Leo Hsu and Regina Obe at Postgres Online for pointing out oidvectortypes. I wrote similar functions before, but used complex nested expressions that this function gets rid of the need for.

See related answer.

(edit in 2016)

Summarizing typical report options:

-- Compact:
SELECT format('%I.%I(%s)', ns.nspname, p.proname, oidvectortypes(p.proargtypes))

-- With result data type: 
SELECT format(
       '%I.%I(%s)=%s', 
       ns.nspname, p.proname, oidvectortypes(p.proargtypes),
       pg_get_function_result(p.oid)
)

-- With complete argument description: 
SELECT format('%I.%I(%s)', ns.nspname, p.proname, pg_get_function_arguments(p.oid))

-- ... and mixing it.

-- All with the same FROM clause:
FROM pg_proc p INNER JOIN pg_namespace ns ON (p.pronamespace = ns.oid)
WHERE ns.nspname = 'my_namespace';

NOTICE: use p.proname||'_'||p.oid AS specific_name to obtain unique names, or to JOIN with information_schema tables — see routines and parameters at @RuddZwolinski's answer.

The function's OID (see pg_catalog.pg_proc) and the function's specific_name (see information_schema.routines) are the main reference options to functions. Below, some useful functions in reporting and other contexts.

--- --- --- --- ---
--- Useful overloads: 

CREATE FUNCTION oidvectortypes(p_oid int) RETURNS text AS $$
    SELECT oidvectortypes(proargtypes) FROM pg_proc WHERE oid=$1;
$$ LANGUAGE SQL IMMUTABLE;

CREATE FUNCTION oidvectortypes(p_specific_name text) RETURNS text AS $$
    -- Extract OID from specific_name and use it in oidvectortypes(oid).
    SELECT oidvectortypes(proargtypes) 
    FROM pg_proc WHERE oid=regexp_replace($1, '^.+?([^_]+)$', '\1')::int;
$$ LANGUAGE SQL IMMUTABLE;

CREATE FUNCTION pg_get_function_arguments(p_specific_name text) RETURNS text AS $$
    -- Extract OID from specific_name and use it in pg_get_function_arguments.
    SELECT pg_get_function_arguments(regexp_replace($1, '^.+?([^_]+)$', '\1')::int)
$$ LANGUAGE SQL IMMUTABLE;

--- --- --- --- ---
--- User customization: 

CREATE FUNCTION pg_get_function_arguments2(p_specific_name text) RETURNS text AS $$
    -- Example of "special layout" version.
    SELECT trim(array_agg( op||'-'||dt )::text,'{}') 
    FROM (
        SELECT data_type::text as dt, ordinal_position as op
        FROM information_schema.parameters 
        WHERE specific_name = p_specific_name 
        ORDER BY ordinal_position
    ) t
$$ LANGUAGE SQL IMMUTABLE;

Run below SQL query to create a view which will show all functions:

CREATE OR REPLACE VIEW show_functions AS
    SELECT routine_name FROM information_schema.routines 
        WHERE routine_type='FUNCTION' AND specific_schema='public';

Get List of function_schema and function_name...

SELECT
    n.nspname AS function_schema,
    p.proname AS function_name
FROM
    pg_proc p
    LEFT JOIN pg_namespace n ON p.pronamespace = n.oid
WHERE
    n.nspname NOT IN ('pg_catalog', 'information_schema')
ORDER BY
    function_schema,
    function_name;

Is a good idea named the functions with commun alias on the first words for filtre the name with LIKE Example with public schema in Postgresql 9.4, be sure to replace with his scheme

SELECT routine_name 
FROM information_schema.routines 
WHERE routine_type='FUNCTION' 
  AND specific_schema='public'
  AND routine_name LIKE 'aliasmyfunctions%';

Example:

perfdb-# \df information_schema.*;

List of functions
        Schema      |        Name        | Result data type | Argument data types |  Type  
 information_schema | _pg_char_max_length   | integer | typid oid, typmod integer | normal
 information_schema | _pg_char_octet_length | integer | typid oid, typmod integer | normal
 information_schema | _pg_datetime_precision| integer | typid oid, typmod integer | normal
 .....
 information_schema | _pg_numeric_scale     | integer | typid oid, typmod integer | normal
 information_schema | _pg_truetypid         | oid     | pg_attribute, pg_type     | normal
 information_schema | _pg_truetypmod        | integer | pg_attribute, pg_type     | normal
(11 rows)

This function returns all user defined routines in current database.

SELECT pg_get_functiondef(p.oid) FROM pg_proc p
INNER JOIN pg_namespace ns ON p.pronamespace = ns.oid
WHERE ns.nspname = 'public';