Real Complex Analysis - BU 2021 Question Paper

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AW-347

B.Sc. III^rd Year (Reg./Pvt.) Main Examination, 2021

Mathematics

Paper - II

Real Complex Analysis

[Maximum Marks: Reg. 40

Pvt. 50

नाष्ट - प्रत्येक खण्ड से पूछ गये प्रशा के अंकस्मान है । प्रत्येक खण्ड का उत्त्तारनवी न पृष्ठ से प्रारम्भा करे हल करें । उत्त्तरपुस्तिका की संख्या 6 सपृम्पठन स्वतःअधि क न हा । विद्याथी द्वारा स्वयं की हस्तलिपिमे उत्त्तार है । विद्याथी उत्तारपुस्तिका का प्रथम पृष्ठ विश्वविद्यालक्की कस.कृटेसे प्राप्त करें । ( प्रत्येकप्रसका उ समान 250 शब् दा से अधिक नहा।)

Note :- All questions from each section carry equal marks. All questions are compulsory and answer limit are approximately 250 words. Start the answer of each section from new page. Maximum limit of pages of answer booklet are approximately 16 pages. Answer should be written by the student in his/her own handwriting mandatory. The first page of answersheet should be download by the student from university website is mandatory.

1.

Let f be a bounded real valued function on [a, b] and p is any portion of [a, b], then prove that:

L [p, f] ≤ U [p, f]

यदि f अन् तरा [a, b] पर वास्तविक मा न परिबद्धफ लन है तब अन् मृस, क] के किस विभाजके लिए सिद्धकी जिए:

L [p, f] ≤ U [p, f]
2.

Test the convergence of

$\int_{-\infty}^{\infty} \frac{dx}{1+x^2}$

अभिभा सरिता का परीक्षक्षण की जिए

$\int_{-\infty}^{\infty} \frac{dx}{1+x^2}$
3.

Let (X, d) be a metric space and let d* be defined by

d* (x,y) = $\frac{d(x,y)}{1+d(x,y)}$ ∀x, y ∈ X

show that d* is a metric on x.

माना(X, d) एक दूरीक समष्टि है निर नप्र का रसे परिभाषित है

d* (x,y) = $\frac{d(x,y)}{1+d(x,y)}$ ∀x, y ∈ X

दशा इए कि x पर एक दूरीक है ।
4.

Define continuous function.

संतत् फलन का परिभाषित की जिए।
5.

Find the moduli and arguments of

मापा कएवं का णा कज्ञातकी जिए

$\frac{3-i}{(2+i)} + \frac{3+i}{(2-i)}$