8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 1/21

RIEMANN 

 Azreen Syazwani Mohd Azmi Nuramalina bte Fauzi 

Hani Azaitie binti Saad

Wan Raihanah binti Meor Idris

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 2/21

Bernhard Riemann• Georg Friedrich Bernhard Riemann.

• Born:17 September 1826 • Died: 20 July 1866 (age 39)

•

German mathematician • His father Friedrich Riemann was Lutheran minister.

• His mother is Charlotte Ebell 

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 3/21

Early years…. 

• Riemann was the second of six children(2

boys and 4 girls), shy, and suffered fromnumerous nervous breakdown.

• His father acted as teacher to his children 

and taught Bernhard until he was 10 years old.

• He was exhibited exceptional mathematical skills, such as fantastic calculation abilities,

from an early age but suffered from timidity and fear of speaking in public.

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 4/21

• 1840-Middle school – lyceum.

• High school – Johanneum Luneberg 

• Study the Bible intensively but often distracted by mathematics.

• 1846, at age 19-started to study philology and theology in order to become a priest and help family’s finances. 

• 1846 – University of Gottingen planning to study towards degree in Theology. However turn to study mathematics under Carl Friedrich Gauss(his lectures is method of least square)

• 1847- transfer to University of Berlin. His lecturers on that time is Dirichlet, Steiner and Einstein.

• He takes two years and returned to Gottingen in 1849.

Education

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 5/21

 

• 1854- held his first lecture which found Riemannian Geometry and set stage for Einstein’s general theory of relativity.

• 1857- attempt to promote Riemann to professor status at University of Gottingen-however fail.

• 1859- after death of Drichlet he was promoted to head mathematics department at Gottingen.

•

1862- married Elise Koch and had a daughter 

 Academia… 

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 6/21

REIMANN’S

CONTRIBUTION

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 7/21

LIST OF REIMANN’S CONTRIBUTIONs INSEVERAL AREAS… 

Combining

analysiswith geometry

Riemannian geometry 

Algebraic geometry 

Complex manifold geometry 

Riemann surface 

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 8/21

Real

analysis

Trigonometric series

Riemann integral = Riemann Sum

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 9/21

 Analyticnumber theory

Riemann zeta function 

Riemann hypothesis 

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 10/21

Complexanalysis

Cauchy – Riemann 

equations 

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 11/21

Sumbangan Riemann

Riemann Sum

-satu proses atau kaedah untuk mencari nilai hampir jumlah luas kawasan di

bawah lengkung pada graf menggunakan subselang.

Terdapat empat jenis Riemann Sum iaitu:

1. Kiri

2. Kanan

3. Titik tengah4. Trapezoidal rule

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 12/21

Sumbangan Riemann

Riemann Sum Kiri

Left (n) = ( f ( x 0) +  f ( x 1) +  f ( x 2) + ... +  f ( xn − 1))Δ x  

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 13/21

Sumbangan Riemann

Riemann Sum Kanan.

Right (n) = ( f ( x 1) +  f ( x 2) +  f ( x 3) + ... +  f ( xn))Δ x  

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 14/21

Sumbangan Riemann

Riemann Sum Titik Tengah

 x x x

 f  x x

 f  x x

 f  x x

 f n Mid  nn

))

2

(...)

2

()

2

()

2

(()( 1322110

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 15/21

Sumbangan RiemannRiemann Sum Trapezoidal Rule

))()(2...)(2)(2)((

2

)(1210 nn x f  x f  x f  x f  x f 

n

abnTrap

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 16/21

 Cauchy–Riemann differential equations 

C onsist of a system of two partial 

differential equations which must be satisfied if we know that a complex function is complex differentiable 

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 17/21

the equations are necessary and sufficient conditions for complex 

differentiation once we assume that its real and imaginary parts are 

differentiable real functions of two 

variables 

Cauchy–Riemann differential equations 

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 18/21

• The Cauchy –Riemann equations on a pair of real-valued functions of two real variables 

u ( x , y  ) and v ( x , y  ) are the two equations:

Cauchy–Riemann differential equations 

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 19/21

• Typically u and v are taken to be the real and imaginary parts respectively of a complex -valued function of a single complex variable z=x+iy , f ( x + i  y  ) = u ( x , y  ) + i v ( x,y  )

Cauchy–Riemann differential equations 

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 20/21

Then f = u + i v is complex-differentiable at that point if and only if the partial derivatives of u and v satisfy the Cauchy –Riemann equations 

(1a) and (1b) at that point 

Cauchy–Riemann differential equations 

8/3/2019 pp riemann

http://slidepdf.com/reader/full/pp-riemann 21/21