Assignment 6 - prl.res.in jayesh/ آ  Assignment 6 Group 5: Apurv & Sanjay Question 2 i. If sin(...

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Transcript of Assignment 6 - prl.res.in jayesh/ آ  Assignment 6 Group 5: Apurv & Sanjay Question 2 i. If sin(...

  • Assignment 6

    Group 5: Apurv & Sanjay

    Question

    2 i. If sin(θ + iφ) = ρ(cosα + i sinα), prove that ρ2 = 1/2[cosh 2φ− cos 2θ]

    Solution

    ρ(cosα + i sinα) = sin (θ + iφ) (0.0.1)

    = sin θ cos iφ+ cos θ sin iφ (0.0.2)

    = sin θ coshφ+ i cos θ sinhφ (0.0.3)

    For two complex numbers to be equal one of the necessary condition is that their magni-

    tude must be equal. Then,

    ρ2(cos2 α + sin2 α) = sin2 θ cosh2 φ+ cos2 θ sinh2 φ (0.0.4)

    ρ2 = (1− cos2 θ) cosh2 φ+ cos2 θ(1− cosh2 φ) (0.0.5)

    = cosh2 φ− cos2 θ (0.0.6)

    = 1 + cosh 2φ

    2 − 1 + cos 2θ

    2 (0.0.7)

    = 1

    2 [cosh 2φ− cos 2θ] Q.E.D (0.0.8)

    1

  • ASSIGNMENT-6

    Group-10(Bivin Geo George and Ritwik Mondal)

    Question 7:

    If

    , then prove that

    Answer:

    Now,

    Using the equation,

    And similarly using

    Therefore from previous,

    Then take the conjugate of that,

    Multiplying the two last equations we got that

  • That’s what it was in the question.

  • P1_G2.pdf P2_i_G5.pdf P2_ii_G4.pdf P3_i_G3.pdf P3_ii_G6_REMARK.pdf P4_i_G7.pdf P4_ii_G11.pdf P5_G8.pdf P6_G9.pdf P7_G10.pdf P8_G1.pdf