Adding function name for ColdStart metric

Author: leandrodamascena

docs/core/metrics.md

@@ -223,6 +223,12 @@ The priority of the `function_name` dimension value is defined as:
 2. `POWERTOOLS_METRICS_FUNCTION_NAME` environment variable
 3. `context.function_name` property
 
+=== "working_with_custom_cold_start_function_name.py"
+
+    ```python hl_lines="4"
+    --8<-- "examples/metrics/src/working_with_custom_cold_start_function_name.py"
+    ```
+
 ### Environment variables
 
 The following environment variable is available to configure Metrics at a global scope:

examples/metrics/src/working_with_custom_cold_start_function_name.py

@@ -0,0 +1,8 @@
+from aws_lambda_powertools import Metrics
+from aws_lambda_powertools.utilities.typing import LambdaContext
+
+metrics = Metrics(function_name="my-function-name")
+
+
+@metrics.log_metrics(capture_cold_start_metric=True)
+def lambda_handler(event: dict, context: LambdaContext): ...