Date post: | 06-Apr-2018 |
Category: |
Documents |
Upload: | azreenazmi |
View: | 227 times |
Download: | 0 times |
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