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γ

को {z ∈ ∶ |z - 1| = 1} से दिए जाने वाले सम्मिश्र समतल में धनात्मक रूप से अभिविन्यस्त वृत्त मान लें। तब निम्न के बराबर है

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  1. 3
  2. 1/3
  3. 2
  4. 1/2

Answer (Detailed Solution Below)

Option 2 : 1/3
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संप्रत्यय:

कॉची समाकल प्रमेय:

यदि एक सम्मिश्र फलन f(z) एक सरल-संबद्ध डोमेन के अंदर एक बंद कंटूर C के अंदर और उस पर विश्लेषणात्मक है, और यदि a, C के मध्य में कोई बिंदु है, तो

f(a) =

व्याख्या:

=

इसलिए ध्रुव दिए गए हैं

(z - 1)(z2 + z +1) = 0 ⇒ z = 1, z = =

γ के अंदर ध्रुव z = 1 है

अतः f(z) = γ के अंदर विश्लेषणात्मक है

इसलिए कॉची समाकल प्रमेय द्वारा,

= f(1) = = 1/3

विकल्प (2) सही है

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