FY B. Tech. Semester II. Complex Numbers and Calculus

FY B. Tech. Semester II Comple Numbers and Calculus Course Code FYT Course Comple numbers and Calculus (CNC) Prepared by S M Mali Date 6//7 Prerequisites Basic knowledge of results from Algebra. Knowledge of Derivatives. Knowledge of Definite and Indefinite integration. Basic knowledge Geometry and Trigonometry. Course Outcomes At the end of the course the students should be able to: CO Find roots of Comple numbers and relate circular functions and hyperbolic functions CO Obtain real and imaginary parts of a comple number. CO Discuss convergence of a series. CO Solve improper integrals. CO Solve differential equation of first order and first degree. CO 6 Epand the given function in powers of and ( a) and evaluate limits. Mapping of COs with POs POs COs CO CO CO CO CO CO 6 a b c D e f g h i j k l m n o Course Contents Unit No. Comple Numbers Title No. of Hours I. Introduction, Modulus and argument of a Comple Number.. Types of Comple numbers.. Algebra of Comple numbers. De Moivre's Theorem (Without proof). Roots of comple numbers 6 Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page

by using De Moivre's Theorem. Epansion of sin(nθ) and cos(nθ) in powers of sinθ and /or cosθ. 6. Epansion of Sin n and Cos n in terms of sines and cosines of multiples of. II Hyperbolic Functions. Hyperbolic Functions. Definitions. Introduction.. Relation between Circular & Hyperbolic functions.. Formulae of Hyperbolic Functions (without proof).. Inverse hyperbolic functions.. Separation of a comple number into real and imaginary parts. 6. Logarithmic function of a comple variable. 6 III. Infinite series. Sequence, series and properties of series.. Series of positive terms, comparison test, integral test.. Ratio test, D Alembert s ratio test.. Root test, Cauchy root test.. Alternating series. 6. Series of positive and negative terms. 6 IV. Improper Integral and special functions. Introduction to improper Integrals.. Improper integral of first and second kind.. Gamma functions and its properties.. Beta functions and its properties.. Relation between Beta and Gamma functions. 6 V. Ordinary Differential Equations of First Order and First Degree:. Definition, order and degree of a differential equation.. Solution of DE of first order and first degree : Linear DE.. Solution of DE reducible to Linear differential equations.. Eact DE and DE Reducible to Eact differential equations.. Applications of DE to orthogonal trajectories. 6. Applications of DE to simple electrical circuits. 6 Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page

VI. Epansion of Functions:. Meaning of epansion of functions. Epansion of a function by Maclaurians series.standard Epansions. Integration, derivative by substitution method.taylor s Series 6.Indeterminate forms 6 Reference Books: Sr. No. Title of Book Author Publisher/Edition Topics 6 Higher Engineering Mathematics. Advanced Modern Engineering Mathematics A tetbook of Engineering Mathematics. Higher Engineering Mathematics. Advanced Engineering Mathematics. Higher Engineering Mathematics. Dr. B. S. Grewal. Glyn James. N. P. Bali, Manish Goyal. H. K. Dass, S. and Er. Rajneesh Verma Chand. Peter V. and O Neil. Ramana B. V. Khanna Publications, Delhi. st edition. Pearson Education (). rd edition. Lami Publications (P) Ltd., New Delhi (). 8 th edition. S. Chand & Company Ltd., () New Delhi. Cengage learning, (). 7 th edition. Tata McGraw Hill Publishing Company, New Delhi, (8). All All All All All All Evaluation scheme Lectures Tutorials Practical Credits -- Evaluation Scheme Component Eam WT Pass FET Theory CAT-I () CAT-II Min ESE Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page

Scheme of Marks Unit No. Title Marks Comple Numbers. 6 Hyperbolic Functions. 8 Infinite Series. 6 Improper Integral and Special functions. 6 Differential Equations of first order and first degree. 8 6 Epansion of functions. 6 Course Unitization Unit Course No. of Questions in Title No Outcomes CAT-I CAT-II Comple Numbers. CO Q. Marks Hyperbolic Functions. CO Q. Marks Infinite Series. CO Q. Marks Improper Integral and Special functions. CO Q. Marks Differential Equations of first order and first degree. CO Q.. Marks 6 Epansion of functions. CO 6 Q.. Marks Unit wise Lesson Plan Unit No Unit Title Comple Numbers Planned Hrs. 6 Lesson schedule Class No. Details to be covered Introduction of comple number, modulus, Argument and Algebra of comple numbers. Statement of De Moivers theorem and eamples. Roots of comle number by using De Moivers theorem. Epansion of sinnѳ, cosnѳ and tannѳ in powers of sinѳ, cosѳ and tanѳ. Definition of circular functions in comple variable. 6 Logarithm of comple number. Review Questions Q Simplify.. cos isin cos isin cos isin cos isin cos isin cos7 isin 7 9 cos isin cos isin 7 6 Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page

Q sin i cos sin i cos Show that. n n cos isin cos n isin n cos isin 8 8. i i sin 6. cos cos 6cos sin tan tan tan tan tan tan Q Epress cos7 and sin 6 in terms of powers of cos and sin Q Q Q6 Q7 Solve..... 9 6 i 7 6 ( ) Find the continued product of all the values of Find all the values of /. i / Find nth root of unity and show that. Roots are in geometric progression. Sum of the all roots is zero Product of all roots is n Q8 Find the common roots of and Q.9. Prove that cos z = cos z Q.. Prove that Sin - z = - i log(iz ± ( - ) ) 8 6 i i Unit No Unit Title Hyperbolic Functions Planned Hrs. 6 Lesson schedule Class Details to be covered No. Introduction of Hyperbolic functions and its properties / Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page

Relation between circular and hyperbolic functions Inverse Hyperbolic functions Separation of a comple number into real and imaginary parts Review Questions Q.. Prove that sinh (i) = i sin and sin(i) = i sinh Q.. Prove that sinh () = sinh cosh and cosh sinh = Q.. Prove that sinh z = sinh sin y and that cosh z = sinh cos y where z = iy Q.. Prove that tanh = tanh tanh tanh y cosh B Q.. If sin (A ib) = iy prove that sinh B = Q.6. Prove that tanh is a periodic function with the period = i Q.7. If tanh = ½, find the value of and sinh Q.8. Find the value of tanh log if = Q.9 Solve the equation 7cosh 8sinh for real values of Q. Prove that Q Q. tanh cosh 6 sinh 6 tanh. sinh sinh 7 7sinh sinh sinh... 7 6 sinh z log( z z ) cosh z log( z z ) z tanh z log z sinh tanh tanh (sin ) cosh (sec ) 6. 7. 8. coth log a a a sech sin log cot If tan tanh Separate into real and imaginary parts i i. i i. tanh iy, prove that. sinh u tan. cosh u sec i. tan e. cos i Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page 6

Q sin( ) If cos( i ) r(cos isin ) then prove that log sin( ) Q If sin( i ) tan isec, prove that coscosh Q tan sin If tan( i) sin( iy), prove that tanh y sinh Q6 If u iv cos ec i, prove that ( u v ) ( u v ) Q7 If iy tan i 6, prove that y cosh iy cosh ( iy) cosh a then prove that Q8 If ( a ) ( a ) y a Q9 If tan( i) i and,y are real prove that is indeterminate and y is infinite Unit No Unit Title Infinite Series Planned Hrs. 6 Lesson schedule Class Details to be covered No.. Sequence, series and properties of series.. Series of positive terms, comparison test, integral test.. Ratio test, D Alembert s ratio test.. Root test, Cauchy root test.. Alternating series. 6 6. Series of positive and negative terms. Review Questions Q. Discuss the convergence of () 8 () () Q.. Use comparison test to show that the series () is divergent Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page 7

()!!! is divergent Q.. Use D Alembert s Ratio test to show that the series () () e e is divergent e is convergent Q.. Use Cauchy root test to show that the series () () () n= n n n n is convergent is convergent is convergent if <, divergent if Q.. Use Leibnitz test to discuss the convergence of the alternating series () () () 6 n (n) n Q.6. Eamine whether the following series are absolutely convergent, conditionally convergent or divergent. ()! ()!!! Unit No Class Unit Title Special Functions Planned Hrs. 7 Details to be covered Lesson schedule Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page 8

No. Improper integral of first and second kind and eamples on it. Gamma function and its properties Beta function and its properties Eamples on Gamma and Beta function Relation between Gamma functions and Beta functions 6 Eamples on Relation between Gamma functions and Beta functions Review Questions Q. Evaluate the following integral. Q.. Q.. Determine if the following integrals are convergent or divergent and if convergent find their value. ➁ Determine if the following integral are convergent or divergent. If convergent find their values. ➁ Q Q Q6 Q7 Q8 Q9 Prove that Γ(n) = nγn Evaluate the following integrals d d b) c) log Evaluate the following integrals ( ) d b) a) c) m n ( ) ( ) d / n a e d d) d sin d) ( )(7 ) Show that Show that Prove that 8 6 ( ) d ( ) / m n 7 / d sin cos d mn m n ( asin bcos ) a b / / d sin sin d B( m, n) a d Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page 9

Q Q Q Prove that Evaluate Prove that d e e a d a b a d log log b ; a >, b > Unit No Unit Title Differential equations of first order and first degree Lesson schedule Class Details to be covered No. Differential equation, Degree, Order and types of solutions Eact Differential equation, Reducible to eact ( Rule,) Reducible to eact ( Rule,), Eamples Linear Differential Equation Non-linear Differential Equation 6 Eamples Review Questions Q Eplain Degree, order of differential equation Q Solve Q Solve y d y dy (sin.cos y e ) d cos.sin y tan y dy Q Solve y a d y b ydy Q Solve a y y d y dy y Q6 Solve e dy y sin d Q7 Solve Q8 Q9 y cos y d log sin y dy dy y Solve d ylog y y Solve yd dy d ( y) y Q Solve Q Solve y y d dy ( e y y ) d a y y dy Q y Solve e ( d dy) e d ye dy Q dy Solve y ( ) d Planned Hrs. 6 Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page

Unit No 6 Unit Title Epansion of functions Planned Hrs. 6 Lesson schedule Class No. Details to be covered Meaning of epansion of functions. Epansion of a function by Maclaurians series. Standard Epansions. Integration, derivative by substitution method. Taylor s Series. 6 Indeterminate forms. Review Questions Q Epand in powers of. tan. log(e ). e sec. tan. log tan 6. 7 6 ( ) ( ) ( ) in powers of by Taylor s theorem Q 7. tan in powers of 8. log( sin ) by Maclaurin s series 9. by using standard epansions e. and hence find f (/) in powers of 7 in powers - Prove that. 6 logsec.... sec.... cos e.... ( )... 6. / log[log( ) ]... 8 Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page

Q Q Q Q6 e log( )... 6. If y y then prove that y... y y y If y... then prove that y...!! Using Taylor s Theorem.. tan(6 6'). sin( '). cos Evaluate sin sin. lim o e sin. lim o log. lim log( ) cot( ). lim tan. lim o sin y y 6. lim y o y 7. lim o 8. lim o e e e sin log tan7 a 9. lim a a. lim. lim o tan sin Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page

Q7. tan lim. lim a sin sin a a a cot( a) lim log a a sin p sin If lim o is finite; find the value of p and the limit Model Question Paper Course Title : FYT : Comple Numbers and Calculus Ma. Marks Duration Hours Instructions: All questions are compulsory Figures to the right indicates full marks Use of non-programmable calculator is allowed Q. Marks No a Find all the values of i / and show that their product is 8 b sin 7 6 Prove that 7 6sin sin 6sin sin Attempt any one of the following 6 c (i) Prove that if the sum and product of two comple numbers are real then either they both must be real or they are comple conjugates. (ii) Prove that the n n th roots of unity are in geometric progression. Attempt any three 8 a If sin( i ) r(cos isin ) then prove that r cos 6 b If sinh cosh find tanh 6 c If sinh θ i φ = e iα, prove that sinh θ = cos α = cos φ 6 d If u = log tan π θ, prove that cosh u = sec θ and tanh u = tan θ 6 a Eamine the convergence of the following infinite series Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page

b Eamine the convergence of the following infinite series e e e c Attempt any one of the following 6 Eamine the convergence of the following infinite series n (n ) n Eamine whether the following series are absolutely convergent, conditionally convergent or divergent.!!!! a Evaluate the following integral. b Determine if the following integral is convergent or divergent and if convergent find its value. c Attempt any one of the following 6 Evaluate the following integrals i) π dθ Sin θ ii) Prove that / / d sin sin d Attempt any three 8 a Solve y a d y b ydy b Solve a y y d y dy y c Solve d e dy y sin d dy d Solve y ( ) Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page

6 a Epand tan in powers of π b y y y If y... then prove that y...!! c Attempt any one of the following 6 Evaluate y y lim y o y Evaluate lim o e e Assignments / Tutorials List of tutorials /assignments to meet the requirements of the syllabus Assignment No. Assignment Title Comple Numbers CO, CO Batch I 7 cos isin cos isin.. Simplify 6 cos isin cos isin.. Show that n cos isin cos n isin n cos isin.. Find all the values of.. Solve.. Solve i 9. 6. Solve for and note all five roots. 7. Prove that / show that their product is sin 7 6 7 6sin sin 6sin sin. 8. If cos cos cos, sin sin sin then show that cos cos cos, sin sin sin Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page

Batch II.. Simplify cos isin cos7 isin 7 9 cos isin cos isin 8 8.. Show that i i 8.. Find the continued product of all the values of i /.. Solve 6 ( ).. Solve. 6. Solve for and note all roots. 7. Prove that 9 tan tan tan tan tan tan. 8. If cos cos cos, sin sin sin then show that Batch III cos cos cos, sin sin sin..simplify sin i cos sin i cos n.. Show that tan tan tan tan tan tan.. Find the continued product of all the values of.. Solve.. Solve 6 i 7 i /. 6. Solve for and note all five roots. 7. Prove that i i. 8. Find nth root of unity and show that. Roots are in geometric progression. Sum of the all roots is zero. Product of all roots is n Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page 6

Assignment No. Assignment Title Hyperbolic Functions CO, CO Batch I ) If z i and n is an integer, then show that z not multiple of ) Define cosh & sinh. Also prove that ) Prove cosh log( ) ) If z if n is n n n n cosh sinh sin( ) cos i R(cos isin ) then prove that log sin( ) ) If tan i iy then prove that 6 6) If tan i i e then prove that y 7) Separate into real and imaginary parts of i) tan i e Prove that Batch II. Solve the equation 7cosh 8sinh Batch III n and log tan tanh (sin ) cosh (sec ) for real values of. Prove that cosh z log( z z ). If tan tanh, prove that. sinh u tan. cosh u sec. Separate into real and imaginary parts. If iy tan i 6, prove that y 6. If sin( i ) tan isec, prove that coscosh. Prove that. Prove that. Prove that Prove that tanh cosh 6 sinh 6 tanh sinh tanh tanh (sin ) cosh (sec ). Separate into real and imaginary parts i i sech sin log cot cos i Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page 7

sin( ). f cos( i ) r(cos isin ) then prove that log sin( ) 6. If u iv cos ec i, prove that ( u v ) ( u v ) cosh 7. If iy cosh ( iy) cosh a then prove that ( a ) ( a ) y a Assignment No Assignment Title Infinite Series CO Batch I Q. Discuss the convergence of A. 8 B. C. Q. Use comparison test to show that the series A. B.!!! is divergent is divergent Q. Use D Alembert s Ratio test to show that the series A. B. e e is divergent e is convergent Q..Use Cauchy root test to show that the series A. B. C. n= n n n n is convergent is convergent is convergent if <, divergent if Q.Use Leibnitz test to discuss the convergence of the alternating series () Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page 8

() 6 Q.6 Eamine whether the following series are absolutely convergent, conditionally convergent or divergent. )! )!!! Batch II Q. Discuss the convergence of A. B. n C. Sin n n Q. Use comparison test to show that the series A. 6 is convergent B. is divergent Q. Use D Alembert s Ratio test to show that the series A. C. 6 is divergent is convergent if Q..Use Cauchy root test to show that the series A. 7 9 is convergent B. 9 6 C. n= a n n n is convergent if < is convergent if <, a < Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page 9

Batch III Q. Discuss the convergence of A. B. C. n n n n! Q. Use comparison test to show that the series A. B. n is divergent 6 6 7 8 is convergent Q. Use D Alembert s Ratio test to show that the series A. B.!! is divergent 6 and the test fails when = > is convergent if < Q..Use Cauchy root test to show that the series A. B. n n= n a n n= n is convergent is convergent if a < Q..Use Leibnitz test to show that the series ( ) n n= n n is convergent Assignment No. Assignment title Improper Integral and special functions CO Batch I ) Prove that Γ(n) = nγn ) Evaluate the following integrals Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page

d d n a i) ii) iii) e d iv) log ) Evaluate the following integrals i) ( ) d ii) ( )(7 ) / / d 7 sin d a b a d log log b / d ) Prove that sin d ) Prove that 6) Prove that ; a >, b > 7) Verify the rule of differentiation under integral sign for the integral a tan d a 8) Define error function and state and prove any two properties of error function a d Batch II ) Evaluate the following integrals a) n a e d d) ) Evaluate the following integrals ) / d sin a d e) ) ( )(7 ) 7 / d e h d ) Show that ) Show that ) Evaluate 8 6 ( ) d ( ) / m n sin cos d mn m n ( asin bcos ) a b B( m, n) Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page

e e 6) Prove that a d a b a d log log b ; a >, b > 7) Verify the rule of differentiation under integral sign for the a tan d a integral 8) Define error function and state and prove any two properties of error function Batch III ) Prove that Γ(n) = nγn ) Evaluate the following integrals d n a ) ) e d ) log ) Evaluate the following integrals. ( ) d. a d ) d e h d ) Prove that / / d sin sin d Prove that...7...(n) n n Assignment No. Assignment Title Differential equation of st order & st degree CO Batch I Solve the following differential equations. (sin.cos y e ) d cos.sin y tan y dy y d y dy.. y a d y b ydy a y y d y dy. Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page

e dy y sin d y. Batch II 6. 7. 8. y cos y d log sin y dy dy y d ylog y y yd dy d ( y) 9. y ( e y y ) d a y y dy. y y d dy y. e ( d dy) e d ye dy dy y ( ) d.. dy e e e d. cos y y dy y ( ) y a d dy y tan d Solve the following differential equations..... dy y e d y( y) d ( y y ) dy y ( y ) d log dy a y y d y dy d y dy Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page

6. 7. 8. 9...... dy tan y ( ) e sec y d dy y e y d yd dy d ( y) ( y y ) d y y y dy y y d dy e sec dy dy ylog y y(log y) d dy y d ( ) dy e e e d y y dy y ( ) y a d. ( y cos y) d (sin ) dy dy y y d y Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page

Batch III Solve the following differential equations. dy y e d.... 6. 7. 8. 9.. y( y) d ( y y ) dy a y y d y dy dy tan y ( ) e sec y d yd dy d ( y) ( y y ) d y y y dy dy ylog y y(log y) d dy y d ( ) dy e e e d y y dy y ( ) y a d. Assignment No 6 Assignment Title Epansion of functions CO6 Batch I. Prove that ( y cos y) d (sin ) dy dy y y d y 7 7 tan. Epand 6 7 8 9 in powers of ( - ) Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page

. Epand. If log log up to y y then epand y in ascending powers of. Find approimate value of tan correct up to four places of decimals 6. Obtain epansion of e sin in powers of up to 7. Using Taylor s theorem epand 8. Evaluate lim o e e sin p sin 9. If lim o Find (i) lim a 6 ( ) in of is finite; find the value of p and limit tan 7 a a (ii) lim Batch II. Epand log e in powers of (-) and hence evaluate log. e correct up to four decimal places. Prove that. Prove that If e log( ) y y then epand y in ascending powers of. Find approimate value of sin '. Obtain epansion of sin e in powers of up to 6. Using Maclaurin s series prove that correct up to four places of decimals 6 sin... e 7. Using Taylor s theorem epand 7 6( ) ( ) ( ) ( ) in powers of 8. Evaluate following limits. lim o e sin log. lim log( ) cot( ) Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page 6

Batch III. lim a. Prove that a tan a. lim o e ( ) log( ) 7 tan n... 7. Epand log e in powers of (-) and hence evaluate loge. correct up to four decimal places. Prove that e log( ). Prove that log log... 8. Find approimate value of sin ' 6. Using Maclaurin s series prove that 7. Using Taylor s theorem epand powers of 8. Evaluate following limits correct up to four places of decimals 6 sin... e 7 ( ) ( ) ( ) ( ) in ) tan lim sin p sin If lim o sin ) lim a sin a a a ) lim log a a is finite; find the value of p and limit cot( a) List of Tutorials - At the end of the tutorial students should be able to: T Solve Eamples on De Moivre s theorem. Find roots of comple numbers. T Obtain real & imaginary parts of a comple number. T Relate circular & hyperbolic functions T Solve eamples on, hyperbolic functions & inverse hyperbolic functions T Discuss convergence of series and use various tests of convergence. T6 Use Cauchy root test for deciding the convergence of alternating series. T7 Solve improper integral and use Gamma and beta functions. Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page 7

T8 T9 T Solve ODE of st order and st degree and apply the knowledge for orthogonal trajectories Epand given functions as power series. Evaluate of indeterminate forms List of open ended eperiments/assignments/ activities Assignment. Solve above given assignments by using scilab and verify your answer. Trace the curves mentioned in the curriculum by using software s like function plotter Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT (CNC) Page 8