I have just started learning Korean for about one month. Now I would like to study Korean grammar in detail.
Could anyone recommend some books that analyze Korean grammar thoroughly? Maybe books that even analyze grammars in greeting language too.
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My suggestion is Korean Grammar In Use series, you have to start with book 1 (beginning to early intermediate).
This book series is written in English, but the examples and practices are in Korean. Each lesson begins with some example sentences, some explanations about the grammatical point, then some example sentences used in conversations, and then ends with some practices. You can easily check your answers with the provided 'answer key' in the appendix.
You can find more explanations about these books here.
You can see the format of the lessons in these sample pictures (from Book 1).
The book that I recommend the most is Korean Grammar for International Learners. This book is published by Yonsei University and I feel is very high quality. I have used this book for about 8 years now. It has served me well as I have progressed from beginner clear through a college degree in Korean. This book does an excellent job at helping you no matter what level your Korean is at. I have seen a lot of Korean grammar books, and this is one of the top choices in my opinion.
Another book that is useful for beginners learning grammar is 500 Basic Korean Verbs. This book (as the title suggests) has 500 of the most used verbs in Korean. It walks you through how to conjugate the verbs in all sorts of tenses and forms. I found this book to be pretty useful, especially with irregular verbs.
You're going to get varied answers for this, but the book used by Ohio State for their Korean program is the Klear Textbook.
The book has 8 lessons. Each 'lesson' has 3 parts(chapters), where each part has 2-3 grammar lessons. This isn't a strict grammar book, but a general textbook to learn Korean (meaning vocabulary too).
I used these series of books+workbooks and they really pack in the grammar into these books, imo.
To your point of grammar in greetings/usages, each chapter has a conversation (+audio recording on their site) as an example that uses the grammar from that lesson.
A resource I find very helpful is The 어미/조사 사전 (전문가용):
This is a dictionary of suffixes, both noun suffixes (조사) and verb endings (어미). Make sure you get the expert version (전문가용), which is yellow; there is also a 초급용 version (for beginners) and 학습자용 version (for learners). The expert version is the biggest, with over 2000 entries. It includes archaic endings too, so if you have any questions about Korean endings you only hear in historical dramas, you can find answers here. Of course there's information about the context and speech level where each ending is appropriate.
Note that it's all in Korean - there's no English. But if you can understand it, I think it's more accurate than resources available in English. There are a lot of examples for each ending, as well.