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RAYAT SHIKSHAN SANSTHA’S S.M.JOSHI COLLEGE HADAPSAR, PUNE-411028

RAYAT SHIKSHAN SANSTHA’S S.M.JOSHI COLLEGE HADAPSAR, PUNE-411028. PRESANTATION BY Prof . DESAI S.S Mathematics department Subject – Complex Analysis Topic –Analytic Functions and C.R Equations. Functions of a Complex Variable. Differentiation of Functions of a Complex Variable.

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RAYAT SHIKSHAN SANSTHA’S S.M.JOSHI COLLEGE HADAPSAR, PUNE-411028

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  1. RAYAT SHIKSHAN SANSTHA’SS.M.JOSHI COLLEGE HADAPSAR, PUNE-411028 PRESANTATION BY Prof . DESAI S.S Mathematics department Subject – Complex Analysis Topic –Analytic Functions and C.R Equations

  2. Functions of a Complex Variable

  3. Differentiation of Functions of a Complex Variable

  4. The Cauchy – Riemann Conditions

  5. The Cauchy – Riemann Conditions (cont.)

  6. The Cauchy – Riemann Conditions (cont.)

  7. The Cauchy – Riemann Conditions (cont.) Hence we have the following equivalent statements:

  8. The Cauchy – Riemann Conditions (cont.) D

  9. Applying the Cauchy – Riemann Conditions

  10. Applying the Cauchy – Riemann Conditions (cont.) D

  11. Applying the Cauchy – Riemann Conditions (cont.)

  12. Differentiation Rules

  13. Differentiation Rules (cont.)

  14. Differentiation Rules

  15. A Theorem Related to z* Iff = f (z,z*) is analytic, then (The function cannot really vary with z*.)

  16. Analytic Functions A function that is analytic everywhere is called “entire”. Composite functions of analytic functions are also analytic. Derivatives of analytic functions are also analytic. (This is proven later.)

  17. THANK YOU

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