[improve](function) add miss proprety in show commands

fe/fe-core/src/main/java/org/apache/doris/catalog/FunctionToSqlConverter.java

@@ -79,6 +79,7 @@ public static String toSql(ScalarFunction fn, boolean ifNotExists) {
             boolean isReturnNull = fn.getNullableMode() == NullableMode.ALWAYS_NULLABLE;
             sb.append(",\n  \"ALWAYS_NULLABLE\"=").append("\"" + isReturnNull + "\"");
             sb.append(",\n  \"VOLATILITY\"=").append("\"" + fn.getVolatility().toSql() + "\"");
+            sb.append(",\n  \"STATIC_LOAD\"=").append("\"" + fn.isStaticLoad() + "\"");
         } else if (fn.getBinaryType() == Function.BinaryType.PYTHON_UDF) {
             appendFileIfPresent(sb, fn, true);
             boolean isReturnNull = fn.getNullableMode() == NullableMode.ALWAYS_NULLABLE;
@@ -164,6 +165,7 @@ public static String toSql(AggregateFunction fn, boolean ifNotExists) {
             boolean isReturnNull = fn.getNullableMode() == NullableMode.ALWAYS_NULLABLE;
             sb.append("\n  \"ALWAYS_NULLABLE\"=").append("\"" + isReturnNull + "\",");
             sb.append("\n  \"VOLATILITY\"=").append("\"" + fn.getVolatility().toSql() + "\",");
+            sb.append("\n  \"STATIC_LOAD\"=").append("\"" + fn.isStaticLoad() + "\",");
         } else if (fn.getBinaryType() == Function.BinaryType.PYTHON_UDF) {
             appendFileIfPresent(sb, fn, false);
             if (fn.getLocation() != null) {

fe/fe-core/src/main/java/org/apache/doris/nereids/trees/plans/commands/ShowFunctionsCommand.java

@@ -287,7 +287,7 @@ private String buildProperties(Function function) {
             properties.put(locationKey, function.getLocation().toString());
         }
         properties.put("NULLABLE_MODE", function.getNullableMode().name());
-        if (function.isStaticLoad()) {
+        if (function.getBinaryType() == Function.BinaryType.JAVA_UDF) {
             properties.put("STATIC_LOAD", String.valueOf(function.isStaticLoad()));
         }
         if (!Strings.isNullOrEmpty(function.getRuntimeVersion())) {

fe/fe-core/src/test/java/org/apache/doris/catalog/FunctionToSqlConverterTest.java

@@ -49,11 +49,25 @@ void testScalarFunction_javaUdf_basicSql() throws AnalysisException {
         Assertions.assertTrue(sql.contains("\"TYPE\"=\"JAVA_UDF\""));
         Assertions.assertTrue(sql.contains("\"VOLATILITY\"=\"volatile\""));
         Assertions.assertTrue(sql.contains("\"ALWAYS_NULLABLE\"="));
+        Assertions.assertTrue(sql.contains("\"STATIC_LOAD\"=\"false\""));
         Assertions.assertFalse(sql.contains("OBJECT_FILE"));
         Assertions.assertFalse(sql.contains("IF NOT EXISTS"));
         Assertions.assertFalse(sql.contains("GLOBAL"));
     }
 
+    @Test
+    void testScalarFunction_javaUdf_staticLoad() {
+        FunctionName name = new FunctionName("testDb", "static_load_fn");
+        Type[] argTypes = {Type.INT};
+        ScalarFunction fn = ScalarFunction.createUdf(BinaryType.JAVA_UDF, name, argTypes,
+                Type.INT, false, null, "com.example.StaticLoadFn", null, null);
+        fn.setStaticLoad(true);
+
+        String sql = FunctionToSqlConverter.toSql(fn, false);
+
+        Assertions.assertTrue(sql.contains("\"STATIC_LOAD\"=\"true\""));
+    }
+
     @Test
     void testScalarFunction_javaUdf_alwaysNullable() {
         FunctionName name = new FunctionName("testDb", "nullable_fn");
@@ -275,13 +289,33 @@ void testAggregateFunction_javaUdf_basicSql() throws AnalysisException {
         Assertions.assertTrue(sql.contains("\"TYPE\"=\"JAVA_UDF\""));
         Assertions.assertTrue(sql.contains("\"VOLATILITY\"=\"stable\""));
         Assertions.assertTrue(sql.contains("\"ALWAYS_NULLABLE\"="));
+        Assertions.assertTrue(sql.contains("\"STATIC_LOAD\"=\"false\""));
         Assertions.assertFalse(sql.contains("INIT_FN"));
         Assertions.assertFalse(sql.contains("UPDATE_FN"));
         Assertions.assertFalse(sql.contains("MERGE_FN"));
         Assertions.assertFalse(sql.contains("IF NOT EXISTS"));
         Assertions.assertFalse(sql.contains("GLOBAL"));
     }
 
+    @Test
+    void testAggregateFunction_javaUdf_staticLoad() {
+        FunctionName name = new FunctionName("testDb", "static_load_sum");
+        Type[] argTypes = {Type.BIGINT};
+        AggregateFunction fn = AggregateFunction.AggregateFunctionBuilder.createUdfBuilder()
+                .name(name)
+                .argsType(argTypes)
+                .retType(Type.BIGINT)
+                .intermediateType(Type.BIGINT)
+                .hasVarArgs(false)
+                .symbolName("com.example.StaticLoadSum")
+                .build();
+        fn.setStaticLoad(true);
+
+        String sql = FunctionToSqlConverter.toSql(fn, false);
+
+        Assertions.assertTrue(sql.contains("\"STATIC_LOAD\"=\"true\""));
+    }
+
     @Test
     void testAggregateFunction_javaUdf_ifNotExists() {
         FunctionName name = new FunctionName("testDb", "my_agg");

fe/fe-core/src/test/java/org/apache/doris/nereids/trees/plans/commands/ShowFunctionsCommandTest.java

@@ -128,6 +128,20 @@ void testBuildProperties_scalarUdfEmitsVolatility() {
 
         Assertions.assertTrue(properties.contains("SYMBOL=com.example.ScalarFn"));
         Assertions.assertTrue(properties.contains("VOLATILITY=immutable"));
+        Assertions.assertTrue(properties.contains("STATIC_LOAD=false"));
+    }
+
+    @Test
+    void testBuildProperties_javaUdfEmitsStaticLoad() {
+        ShowFunctionsCommand sf = new ShowFunctionsCommand("test", true, null);
+        ScalarFunction fn = ScalarFunction.createUdf(Function.BinaryType.JAVA_UDF,
+                new FunctionName("test", "java_static_load_fn"), new Type[] {Type.INT},
+                Type.INT, false, null, "com.example.StaticLoadFn", null, null);
+        fn.setStaticLoad(true);
+
+        String properties = sf.buildPropertiesForTest(fn);
+
+        Assertions.assertTrue(properties.contains("STATIC_LOAD=true"));
     }
 
     @Test