# Tool Use Mathematics and Implementation: From JSON Schema to Pydantic AI — Production Agent Engineering

> Source: https://sukruyusufkaya.com/en/learn/llm-muhendisligi/tool-use-matematik-implementation-pydantic
> Updated: 2026-05-13T13:00:31.880Z
> Category: LLM Mühendisliği
> Module: Module 20: AI Agents — Tool Use, Function Calling, MCP, Multi-Agent
**TLDR:** Internal mathematics of tool use and production implementation: JSON schema standard detail, full anatomy of OpenAI function calling, ReAct prompt engineering techniques, MCP protocol implementation (Python stdio + SSE). Turkish tool calling examples (TC ID validation, e-invoice query). Clean, type-safe agent with Pydantic AI. Modern approach as LangChain alternative. Error handling, retry logic, tool timeout management.